Handbook of Semiconductor Technology.. Electronic Structure and Properties of Semiconductors (Wiley-Vch, 2000)(ISBN 3527298347)(O)(861s)

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Handbook of Semiconductor Technology Volume 1 Kenneth A. Jackson, Wolfgang Schroter (Eds.)

Weinheim . New York Chichestera Brisbane Singapore Toronto a

Editors: Prof. K. A. Jackson The University of Arizona Arizona Materials Laboratory 4715 E. Fort Lowell Road Tucson, AZ 857 12. USA

Prof. Dr. W. Schroter IV. Physikalisches Institut der Georg-August-Universitat Gottingen Bunsenstfafie 13- 15 D-37073 Gottingen. Germany

This book was carefully produced. Nevertheless, authors, editors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: applied for Deutsche Bibliothek Cataloguing-in-Publication-Data A catalogue record is available from Die Deutsche Bibliothek ISBN 3-527-29834-7

0 WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2000 Printed on acid-free and chlorine-free paper. All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition, Printing and Bookbinding: Konrad Triltsch. Print und digitale Medien GmbH, D-97070 Wiirzburg Printed in the Federal Republic of Germany.

Preface

Rapid scientific and technological developments have enabled microelectronics to transform the computer industry of the sixties into today's information technology, which is now revolutionizing communications and the information media. Larger than the car business, information technology is one of the industries most impacted by physical research and technology transfer in the 20thcentury. This will continue for at least the first two decades of the 21" century. The two volumes of this Handbook describe the underlying scientific and technological bases of this unique development, with the first addressing the science, and the second the technological framework of the field. Written by experts who have made major contributions to this enterprise, the chapters span from defect physics to device processing to present a panorama of the key steps, models, and visions - in short the evolution - of microelectronics. At the same time, this Handbook can be seen as a carefully written status report, specially valuable to those engaged in the continuing interplay between semiconductor science, technology, and business, and in the creation of new markets, such as sensor arrays, power and high frequency devices, solar cells, and blue lasers. Circuit and systems design, which turn science and technology into end-user products, are not included as separate chapters, because each would need a volume in its own right. However, due to their close connection with process science and technology, they are briefly treated as the need arises. In semiconductors, science frequently develops in close interplay with technology, and fundamental investigations and technological advances cross-pollinate each other in an unprecedented fashion. The miniaturization of the transistor, begun forty years ago, is approaching dimensions, where present concepts appear to break down, and the available characterization methods may no longer function. Other devices, such as solar cells, are now entering the mass market, placing increasing demand on materials quality, process efficiency, and, of course, cost. At present, the most promising approach to addressing these challenges appears to involve fundamental understanding and modeling of highly complex nonlinear solid state phenomena, in short, physically-based, predictive simulation of complex process technological sequences. The first volume places particular emphasis on the concepts and models relevant to such issues. Starting with a description of the relevant fundamental phenomena, each chapter describes and develops the mechanisms and concepts used in current semiconductor research. Experimental details are provided in the text, or summarized in tables and diagrams to the extent needed to illustrate the models under discussion. The Handbook begins with chapters on the basic concepts of band structure formation, charge transport, and optical excitations (chapters 1 and 2), the physics of defects (point defects, impurities, dislocations, grain boundaries, and interfac-

VI

Preface

es) in crystalline semiconductors, particularly Si and GaAs (chapters 3-8, lo), special materials, such as hydrogenated amorphous Si (chapter 9), concluding with semiconductors for solar cell applications, silicon carbide, and gallium nitride (chapters l l - 13). I am very grateful to the contributors who took the trouble to write a chapter for this volume. I thank Prof. Peter Haasen, Prof. Abbas Ourmazd, PD Dr. Michael Seibt, and Prof. Helmut Feichtinger for many useful proposals and critical comments. I also thank Dr. Jorn Ritterbusch and Mrs. Renate Dotzer of WILEYVCH for their advice and very agreeable cooperation. Let me finally quote from a letter of one of the authors (A. 0.):“We wish the reader as much fun with the material as we have had - as much fun, but much less hard work”.

Wolfgang Schroter Gottingen, April 2000

Contents

1 Band Theory Applied to Semiconductors . . . M. Lannoo 2 Optical Properties and Charge Transport . . . R. G. Ulbrich 3 Intrinsic Point Defects in Semiconductors 1999 G. D. Watkins 4 Deep Centers in Semiconductors H. Feichtinger

.............

1

.............

69

. . . . . . . . . . . . 121

....................

5 Point Defects, Diffusion, and Precipitation I: Y: Tan, U. Gosele 6 Dislocation . . . . . . . . . . . . . . . . . H. Alexander, H. Teichler

167

. . . . . . . . . . . . . . . 23 1 ..............

291

7 Grain Boundaries in Semiconductors . . . . . . . . . . . . . . . . . . 377 J. Thibault, J.-L. Rouviere, A. Bourret 8 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 R. Hull, A. Ourmazd, u!D. Rau, P. Schwandel; M . L. Green, R. T. Tung 9 Material Properties of Hydrogenated Amorphous Silicon R. A. Street, K. Winter

. . . . . . . 541

10 High-Temperature Properties of Transition Elements in Silicon W Schroter, M. Seibt, D. Gilles

11 Fundamental Aspects of S i c W J. Choyke, R. P. Devaty

....

......................

597 661

12 New Materials: Semiconductors for Solar Cells . . . . . . . . . . . . 7 15 H. J. Moller 13 New Materials: Gallium Nitride . . . . . . . . . . . . . . . . . . . . . 771 E. R. Weber, J. Kriiger, C. Kisielowski Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

1 Band Theory Applied to Semiconductors Michel Lannoo DCpartement Institut SupCrieur d’Electronique du Nord. Institut d’Electronique et de Microdectronique du Nord. Villeneuve d’Ascq. France

3 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 From Discrete States to Bands . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 1.1.2 Bloch Theorem for Crystalline Solids . . . . . . . . . . . . . . . . . . . . . 9 1.1.3 The Case of Disordered Systems . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 The Effective Mass Approximation (EMA) . . . . . . . . . . . . . . . . . . 10 1.1.4.1 Derivation of the Effective Mass Approximation for a Single Band . . . . . 10 1.1.4.2 Applications and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 The Calculation of Crystalline Band Structures . . . . . . . . . . . . . . 14 14 1.2.1 Ab Initio Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1.1 The Hartree Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1.3 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . . . 1.2.1.4 Beyond Local Density (the G-W Approximation) . . . . . . . . . . . . . . . 17 1.2.1.5 The Pseudopotential Method . . . . . . . . . . . . . . . . . . . . . . . . . 17 19 1.2.2 Computational Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.2.1 Plane Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2.2 Localized Orbital Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 1.2.3.1 The Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . 22 1.2.3.2 The Empirical Pseudopotential Method . . . . . . . . . . . . . . . . . . . . Comparison WithExperimentsfor ZincBlendeMaterials . . . . . . . . 23 1.3 23 1.3.1 The General Shape of the Bands . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1.1 The Tight Binding Point of View . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.1.2 The Empirical Pseudopotential Method . . . . . . . . . . . . . . . . . . . . 1.3.2 The k-p Description and Effective Masses . . . . . . . . . . . . . . . . . . 29 31 1.3.3 Optical Properties and Excitons . . . . . . . . . . . . . . . . . . . . . . . . Ab Initio Calculations of the Excitonic Spectrum . . . . . . . . . . . . . . . 34 1.3.4 1.3.5 A Detailed Comparison with Experiments . . . . . . . . . . . . . . . . . . 34 1.4 Other Crystalline Materials with Lower Symmetry . . . . . . . . . . . . 36 General Results for Covalent Materials 1.4.1 with Coordination Lower than Four . . . . . . . . . . . . . . . . . . . . . . 36 . . . . . . . . . . . . . . . . . . . . . 37 Chain-Like Structures Like Se and Te 1.4.2 1.4.3 Layer Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.4.4 New Classes of Materials: the Antimony Chalcogenides . . . . . . . . . . . 39

2

1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 1.6.1 1.6.2 1.7 1.7.1 1.7.2 I .7.3 1.7.4 1.8

1 Band Theory Applied to Semiconductors

Non-Crystalline Semiconductors . . . . . . . . . . . . . . . . . . . . . . The Densities of States of Amorphous Semiconductors . . . . . . . . . . . Numerical Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . Dangling Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case of SiO, Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . Disordered Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of the Different Approximations . . . . . . . . . . . . . . . . The Case of Zinc Blende Pseudobinary Alloys . . . . . . . . . . . . . . . Systems with Lower Dimensionality . . . . . . . . . . . . . . . . . . . . Qualitative Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Envelope Function Approximation . . . . . . . . . . . . . . . . . . . . Applications of the Envelope Function Approximation . . . . . . . . . . . Silicon Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

. 44 46 47 49 51 . 52 . 54 57 57 59

. 61 63 65

List of Symbols and Abbreviations

List of Symbols and Abbreviations lattice parameter basis vector of the unit cell bandgap energy conduction-band and valence-band energy electron charge envelope function reciprocal lattice vector resolvent operator hamiltonian Planck constant elements of the hamiltonian total angular momentum joint density of states wave vector orbital angular momentum optical matrix element effective mass density of states momentum vector momentum of electron and hole matrix element of the momentum operator spin vector kinetic energy transfer matrix Bloch's function potential crystal volume perturbation coupling constant splitting energy energy of the state n real and imaginary part of the dielectric constant wave length frequency self-energy operator conductivity atomic states wave function volume of the unit cell atomic volume frequency

3

4

ATA a-Si, c-Si CPA DOS EMA EMT EPM EPR ESR ETB EXAFS HOMO LCAO LDA LUMO MCPA TBA UPS VCA VSEPR XPS


average r matrix approximation amorphous and crystalline silicon coherent potential approximation density of states effective mass approximation effective mass theory empirical pseudopotential method valence pair repulsion electron spin resonance empirical tight binding extended X-ray fine structure highest occupied molecular orbital linear combination of atomic orbitals local density approximation lowest unoccupied molecular orbital molecular coherent potential approximation tight binding approximation ultraviolet photoemission spectroscopy virtual crystal approximation valence shell electron pair repulsion X-ray photoemisssion spectroscopy

1.1 General Principles

1.1 General Principles The aim of this chapter is to establish the basic principles of the formation of bands in solids. To do this we start with a one-dimensional square well potential which we consider as a simplified description of an atom. We then bring two such square wells into close contact to simulate the behavior of a diatomic molecule. This can be generalized to include a large number of square wells to illustrate the concept of energy bands corresponding to the one-dimensional model of the free electron gas. These results are shown to be independent of the boundary conditions, allowing us to use Born von Karmann cyclic conditions and to classify the band states in terms of their wave vectors. Such a simple description does not take into account the spatial variations of the true potential in crystalline solids. However, the free electron gas results can be generalized through the use of Bloch’s theorem which allows the classification of all energy bands in terms of their wave vectors, the periodic boundary conditions imposing that the allowed energy values be infinitely close together. This does not hold true, however, in nonperiodic systems such as amorphous solids and glasses. However, we show that, in this case, although general considerations impose the existence of bands, these can contain two types of states with localised or delocalised behavior. This represents the fundamental difference between nonperiodic systems and crystalline materials. A final situation of practical interest is the case where a slowly varying potential is superimposed on a rapidly varying crystal potential. Such cases can be treated in a simple and very efficient manner by the effective mass theory (EMT), also termed the envelope function approximation. The best known applications of the EMT correspond

5

to hydrogenic impurities in semiconductors and, more recently, to various kinds of system involving semiconductor heterojunctions, which will be discussed in Sec. 1.7.

1.1.1 From Discrete States to Bands We want to describe here in the simplest way the basic phenomena which occur when atoms are brought close to each other to form a solid. To do this, the attraction potential of the free atom is represented by a one-dimensional square well potential and the main trends concerning its energy levels and wave function are analyzed. We first consider the square well potential of Fig. 1 - 1, with depth V , and width a. We assume a to be of atomic dimensions (i.e., a few A) and take V, to be large enough for the energy E of the lowest states to be such that E e V,. In such cases these lowest states are close to those of an infinite potential, Le., their wave functions t+~,, and energies E,, are given by:

h2 -2m

y,, sin k,, x &,

k,,

nn --

( n = 1 , 2 , ...)

U

the allowed values k,, of the one-dimensiona1 wave vector k practically corresponding

V

a

Figure 1-1. One dimensional square well potential of height V, and of width a.

6


to vanishing boundary conditions at x = 0 and a. For a typical atomic dimension of a = 3 p\, the distance in energy between the lowest two levels is of the order of 10 eV, typical of the values found in atoms. Of course the values of the excited states cannot correspond to what happens to true three-dimensional potentials, but this will not affect the main qualitative conclusions derived below. Let us now build the one-dimensional equivalent of a diatomic molecule by considering two such potentials at a distance R (Fig. 1-2). We can discuss qualitatively what happens by considering first the infinitely large R limit where the two cells can be treated independently. In this limit, the energy levels of the whole system are equal to those of each isolated well (i.e., E l , E, . . . ) but with twofold degeneracy since the wave function can be localised on one or the other subsystem. When R is finite but large, the solutions can be obtained using the first order perturbation theory from those corresponding to the independent wells. This means that the twofold degenerate solutions at E , , E, . . . will exhibit a shift and a splitting in energy resulting in two sublevels of symmetric and antisymmetric character. This shift and splitting will increase as the distance R decreases. Such a behavior is pictured in Fig. 1-3, from R = to the limiting case where R = a. This case is particularly easy to deal with since it corresponds to a single potential well but

r

0

R

I

a

Figure 1-2. The double well potential as a simple example of a molecule.

a

Figure 1-3. Energy levels for the double well of Fig. 1-2 as a function of the distance R between the two wells, a being the width of the individual wells.

Figure 1-4. Formation of bands for a Kronig-Penney model where a is the well width and R the interwell distance.

of width 2a instead of a . The consequence is that one again gets the set of solutions Eq. ( 1 - 1 ), but the allowed values of k, are closer together with an interval x l 2 a instead of nla. This means that the number of levels is multiplied by two as is apparent in Fig. 1-3. The generalization to an arbitrary number N of atoms is obvious and is pictured in Fig. 1-4. At large inter-well separation R , the degeneracy of each individual level E , , E, . . . is N . At closer separation, these levels shift and split into N distinct components. When R = a one recovers one single well of width N u , which means that the allowed values of k, are now separated by x l N a . For a crystal, N becomes very large


(or order lo7 for a 1D system) and these allowed values form a “pseudocontinuum”, Le., a set of discrete values extremely close to one another. The same is true of the energy levels whose pseudocontinuums can either extend over the whole range of energies (the case R = a in Fig. 1-4) or more generally ( R > a ) are built from distinct groups of N levels. These groups of N levels are always contained in the same energy intervals irrespective of the value of N . They are called the allowed energy bands, the forbidden regions being called gaps. These qualitative arguments can be readily generalized to realistic atomic potentials in three dimensions. One then gets a set of allowed energy bands which contain a number of states equal to the number of atoms N times an integer. These bands are, in general, separated by forbidden energy gaps.

1.1.2 Bloch Theorem for Crystalline Solids For crystals it is possible to derive fairly general arguments concerning the the properties of the allowed energy bands. Before doing this let us come back to our simplified model of N square wells in the situation R = a where one gets a single well of width N a. We have seen that the states are completely characterized by the allowed values k , of the wave number k which are equal to n xlNa with n > 0. It is to be noted, however, that even this simple model contains cornplications due to the existence of boundaries or surfaces. It is the existence of these boundaries that lead to solutions having the form of stationary waves sin(kx) instead of propagating waves exp(i kx) which are mathematically simpler (the sine function is a combination of two exponentials). Of course the increased mathematical complexity of the vanishing boundary conditions is not very important for this particu-

7

lar one-dimensional problem, but it becomes prohibitive in real cases where one has to deal with the true surfaces of a threedimensional material. If one only needs information about volume properties it is possible to avoid this problem by making use of Von Laue’s theorem which states that perturbations induced by surfaces only extend a few angstroms within the bulk of the material. This means that for large systems, where the ratio of the numbers of surface to volume atoms tends to zero, volume properties can be obtained by any type of boundary conditions even if they do not seem realistic. In that respect, it is best to use Born von Karman’s periodic conditions in which one periodically reproduces the crystal under study in all directions, imposing that the wave function has the same periodicity. One then gets an infinite periodic system whose mathematical solutions are propagating waves. For instance, the linear square well problem with R=a becomes a constant potential extending from minus to plus infinity. Its solutions can be taken as:

E,

A= k,2 =-

2m

2nm k, =- L ( n 0 integer) They are mathematically different from Eq. (1-1) but they lead to the same answer for physical quantities. For instance, the interval between the allowed values of k is larger by a factor of two but this leads to the same density of states (number of levels per unit energy range) because the states are twofold degenerate (states with opposite values of k give the same energy). One can also show that the Fermi energy and electron density are identical in both cases.

8


The results given by Eq. (1 -2) can be generalized to real crystalline solids through the use of the Bloch theorem. Such solids are characterized by a periodically repeated unit cell and, with Born von Karman boundary conditions, they have translational periodicity which greatly simplifies the mathematical formulation of the solutions. Let us then consider a three-dimensional crystal for which one has to solve the one-electron Schrodinger equation:

H v ( r )= E V ( r )

(1 -3)

where r is the electron position vector. If we call aj the basis vectors of the unit cell, the fact that the system has translational symmetry imposes that:

I v (r+aj) I2 = I 11, ( r )I'

(1-4)

which, for the wave function itself, gives:

v (r+aj) = exp (1 q j ) V ( r )

(1-5)

If we now consider a translation by a vector R

R = C m, a, J

mj being integers, we automatically get the condition:

which defines a new function uk (r). The direct application of a translation R to this expression leads to: (1-10) qk(r + R ) = exp [i k ( r + R ) ] uk (I + R ) and a comparison with Eq. (1-8) automatically leads to: uk ( r + R ) = uk ( r )

(1-1 1)

i.e., that the function uk ( r )is periodic. Equations (1-9) and (1-1 1) constitute Bloch's theorem, which states that the eigenfunctions can be classified with respect to their wave vector k and written as the product of a plane wave exp(ikr) times a periodic part. They are thus propagating waves in the crystal lattice. This is a generalization of the one-dimensional case discussed above. If one injects Vk(r)into Schrodinger's equation Eq. (1 -3), then the eigenvalue will become a continuous function E ( k ) of the wave vector k. However, the vector k can only take allowed values chosen in such a way that wk(r)satisfies the boundary conditions, which are: V k (r + Nj

aj) = q k ( r )

(1-12)

where N, is the number of crystal cells along a,. Using Eq. (1 -9) and Eq. (1-1 1) this imposes the conditions:

The phase factor in Eq. (1-7) is a linear function of the components of R and can be written quite generally under the simpler form k R , leading to: a

vk(r + R ) = exp (i k

R ) Vk ( r )

(1-8)

where k is such that k . a, is equal to (in this expression we have indexed the wave function by its wave vector k). To exploit Eq. (1 -8), which is a direct consequence of translational invariance, we can write without loss of generality:

vk( r )= exp (i k

*

r ) uk ( r )

( 1-91

(1-13) where nj is an integer. These allowed values can be expressed more directly with the help of the basis vectors of the reciprocal lattice a; defined by: a,? a , = 2rc

Sj,

(1-14)

which leads to: (1-15)


This again generalizes the one-dimensional situation described by Eq. (1-2) to any real crystal in 1-, 2-, or 3-dimensions. As a conclusion Bloch’s theorem leads us to write the wave function as a plane wave exp (i k r ) modulated by a periodic part. Its energy E ( k ) is a continuous function of the wave vector k . This one takes discrete values which form a pseudocontinuum. The allowed energies will then be grouped into bands as discussed more qualitatively in the previous section. A final point is that one can show that the energy curves E ( k ) and Bloch functions Vk(r)are periodic in reciprocal space. One then gets the complete information about these quantities from their calculation fork points lying in one period of the reciprocal lattice. From the symmetry properties ( E @ ) = E ( - k ) ) it is better to use the period symmetrical with respect to the origin, called the first Brillouin zone.

1.1.3 The Case of Disordered Systems Different varieties of solids exist which do not exhibit the long range order characteristic of perfect crystals. They differ qualitatively among themselves by the nature of their disorder. A simple case is the alloy system where one can find either A or B atoms on the sites of a perfect crystalline lattice. This kind of substitutional disorder is typical of the ternary semiconductor alloys Ga,,AI,As where the disorder only occurs on the cationic sites. We shall deal in more detail with this problem in Sec. 1.6. Other cases correspond to amorphous semiconductors and glasses which are characterized by a short range order. For instance, in a-Si, the silicon atoms retain their normal tetrahedral bonding and bond angles (with some distortions), but there is a loss of long range order. A lot of covalently-bonded systems (with coordination numbers small than 4) can be found in the amorphous or

9

glassy state. All have a short range order and moderate fluctuations in bond length, but in some cases they can have large fluctuations in bond angles. In all these situations, even for the disordered substitional alloy, it is clear that one cannot make use of Bloch’s theorem to classify the band states. We could even ask ourselves if the concept of energy bands still exists. The simplest case demonstrating this point again corresponds to the one-dimensional system with square well potentials. We can simulate a disordered substitutional alloy by considering square wells of width a as before, and of depth V , for A atoms and V, for B atoms, with V,> V,. We also assume that the interwell distance is R = a to get the simplest situation. If the material were purely B then one would get a constant potential which we take as the origin of energies. Thus each time one substitutes an A atom for a B one this results in an extra potential well of width a a depth V,-V,. Two A neighbors lead to a well of width 2 a , and a cluster of M A neighbors gives a well of width M a (Fig. 1-5). The alloy will consist of a distribution of these potential wells separated by variable distances. It is interesting to investigate the nature of the states for such a problem. The simplest case corresponds to energies E > 0, where the states have a propagating behavior and there are solutions at any positive value of E. The existence of boundary conditions due to the fact that the system contains a finite but large number N of atoms

- I- -J a

20

Figure 1-5. Simple representation of a disordered binary alloy with square well potentials of width a for one impurity, 2 a for a pair, etc.

10


will simply transform this continuum of states into a pseudocontinuum. This results in a band of “extended states” for E > 0. The situation for E e 0 is drastically different. An isolated impurity A, represented by a single well, gives at least one energy level at E , , or even more at E,, E,, etc. The corresponding wave function is localized and decays exponentially as exp (- k I z I) 2m where k = - / E l . Clusters of M atoms

1 ti2

lead, as described before, to denser sets of levels in the well since the interval between allowed k values is divided by M . In particular, for a very large cluster, one obtains a pseudocontinuum of levels, the lowest one being infinitely close to the bottom of the potential well. One has thus a statistical distribution of potential wells of varying width giving rise to a corresponding distribution of levels located between E=O and E = - (V,- VB). Again this leads to pseudocontinuum of levels. Furthermore, there is the possibility of interaction between the potential wells when these are close enough. This also acts in favor of a spread in energy levels. The states in the potential wells can have a more or less localized character depending on the distance between the wells. We shall discuss this problem later. The conclusion of this simple model is that the occurrence of disorder, at least in sornes cases, also leads to the existence of a pseudocontinuum of states, even if Bloch’s theorem does not apply. Of course we have only discussed one particular model. We shall later discuss other cases of disorder or randomness which also lead to the existence of well-defined energy bands.

1.1.4 The Effective Mass Approximation @MA) Interesting cases that often occur in practice correspond to the application of a po-

tential slowly varying in space to a crystalline solid. Such situations can be handled relatively easily without solving the full Schrodinger equation, by using the so called “effective mass approximation“. One major field of application has been the understanding of hydrogenic impurities in semiconductors (for reviews see Bassani et al., 1974; Pantelides, 1978). More recently the same method, often called “the envelope function approximation”, has been applied to the treatment of semiconductor heterojunction and superlattices as will be discussed in Sec. 1.7.

1.1.4.1 Derivation of the Effective Mass Approximation for a Single Band Let us begin with the simplest case of a crystal whose electronic structure can be described in terms of a single energy band, the solution of the perfect crystal Schrodinger equation: Ho (pk ( r )= E ( k ) (pk ( r )

(1-16)

If a perturbative potential V ( r )is applied to the system we can describe a solution of the perturbed system I# ( r )as a linear combination of the perfect crystal eigenstates (k belonging to the first Brillouin zone), (1-17) k

and obtain the unknown coefficients by projecting the new Schrodinger equation, (Ho + v)

v ( r )= E I# ( r )

(1-18)

onto the basis states (pk ( r ) .This immediately leads to the set of linear equations:

E ( k ) a ( k ) + x( ( P k l V l ( P P ~ ) a ( k ’ ) = E a ( k ) k‘ (1-19) At this stage we need to simplify the matrix elements of V , otherwise it is impossible to


go further, except numerically. We use the fact that the a ( r )are Bloch functions,

a ( r )= eik

uk(r)

*

( 1-20)

and express the potential matrix element as:

(%WI

(1-21)

(pk’)=

= J~

( ei(k’-k) ~ 1 . r uk(r) * ukg( r )d3 r

In view of the Bloch theorem the product uz uk,is a periodic function of r and we can expand it in a Fourier series,

where v is the crystal volume and G are the reciprocal lattice vectors. The matrix element Eq. (1-21) can thus be expressed exactly as: (1-23 a)

11

However, Cko,ko(0) has an important property: it is given by the following integral over the crystal volume

J- uk*,(r>uko(r)d3 r = = J qzo(r)(pko(r) d3 r = 1

Ck”,k0(O)=

(1 -25)

in view of the fact that the wave functions ~ ( r are ) , normalized. The final form of the EMA equation is thus (1-26) V ( k ’ k ) a ( k ’ ) = Ea@) E ( k )a ( k ) + k’

It is interesting to derive a real space equation from this by Fourier transforming Eq. (1 -26). To perform this we must take into account the fact that the function a ( k ) is strongly peaked near ko. We thus introduce the following Fourier transform:

a ( k ) exp (i ( k - ko ) r )

F ( r )=

(1 -27)

k

1

V ( r )d r

V ( q )= -

(1-23b)

V

At this level we must make some assumptions about V ( r ) .The first one is that it varies slowly in space (i.e., over distances which are large compared to the size of the unit cell). This means that its Fourier transform decreases very rapidly with the modulus of the wave vector, Le., that one can neglect terms with G#O in Eq. (1-23) and, also, that only terms with k ’ = k will effectively contribute. We now make the second central assumption of the EMA, that we look for solutions whose energy E is close to a band extremum k,. If this is so, only states with k = ko will have a ( k ) sensibly different from zero in Eq. (1-19). This means that one can rewrite Eq. (1-19) using Eq. (1-23) under the approximate form: (1-24) E ( k ) a ( k ) + C c , , , , , ( O ) V ( k ’ - k ) a ( k ’ ) =0 k’

such that F(r) varies slowly in space when a ( k ) only takes important values in the vicinity ofk=ko. To get an equation for F ( r ) we multiply Eq. (1-26) by expi(k-ko) r and sum over k , assuming that one makes a negligible error in the potential term by extending the summation over k to the whole space. This leads to the real space equation: (E(ko-i Vr)+V(r)}F ( r ) = E F ( r ) (1-28) This is a differential equation in which the operator k,-iVr has been substituted f o r k in the dispersion relation E ( k ) . As we have seen, the function F ( r ) is likely to vary slowly with r (or a ( k ) # 0 only fork= k,) so that one can expand E ( k ) to the second order in the neighborhood of k ko. Calling a the principal axes of this expansion we have:

-

which defines the effective masses ma along direction a.This allows us to rewrite

12


scaled parameters e’ .--+ e2/&and m +m *. This leads to a set of hydrogenic levels with

Eq. (1 -28) as:

an effective Rydberg

={E-E(ko)J F ( r ) which represents the usual form of the EMA equation as derived by many authors (Bassani et al., 1974; Pantelides, 1978). It is interesting to examine the meaning of the function F(r).To do this we start from the expansion Eq. ( 1 -17) of q ( r ) , express the qk(r)as in Eq. (1-20) and factorize ,i ko r . This gives

2A2 e? typical values of m* = 0.1 and

( 1-32) q ( r )= F ( r ) (pko ( r ) This means that q ( r )can be rewritten as the

product of the Bloch function (which varies over a length typically of the order of the interatomic distances) times a slowly varying “envelope function”. The advantage of the EMA is that one directly obtains F(r) from a Schrodinger-like equation involving the effective masses. 1.1.4.2 Applications and Extensions

The first well known use of the EMA was for hydrogenic impurities in semiconductors. If we treat single donor substitutional impurities, like As in Si or Ge for instance, the excess electron will see an attractive potential roughly given by - e 2 / &r (where E is the dielectric constant), which one considers as slowly varying. This can stabilize levels in the gap in the proximity of the bottom of the conduction band E,. For a single minimum and an isotropic effective mass one gets an hydrogenic-like equation with

which, for

E = 10, becomes of order 14 meV, i.e., fairly small compared to the band gap. This result correctly reproduces the order of magnitude found in experimental data. However, to be truly quantitative, the EMA must in many cases satisfy the following requirements:

-

k

As a ( k ) is peaked near ko, we approximate u k ( r ) by its value at k o , which leads us directly to:

m*e4

___

-

It must include the effective-mass anisotropy when necessary. This has been done in Faulkner (1968, 1969), one effect being the splitting of p states, for instance. It must also properly include the valleyvalley interactions when there are several equivalent minima. This can be done, for instance, by first order perturbation theory on degenerate states (since there are as many identical impurity states than there are minima).

With these improvements the EMA theory has achieved considerable success for single donor impurities, especially for excited states (see Bassani et al., 1974); Pantelides, 1978) and Table 1-1 for reviews. Only the ground state is found to depart significantly from the predicted levels at this stage of the theory. This is due to the deviations of the potential from its idealized form - e 2 / &r in the impurity cell. The corresponding correction is known as the chemical shift. It is also possible to treat more exactly the many-valley interactions by recently derived methods described in Resca and Resta (1 979, 1980). The case of acceptor states derived from the valence band is more complicated. This is due to the threefold degeneracy of the top of the valence band. This means that ( r ) must be written as a combination of the

13


Table 1-1. Comparison between theoretical and experimental energy spacing (cm-’) for donor impurities in silicon. The theoretical values are taken from Faulkner (1968, 1969). (The spacing between excited states, independent of the ground state position, is more suitable than the observed position of the transition from the ground state to perform a comparison with the theory since this ground state is not hydrogenic.) Transition

Theory

P

As

Sb

2 P* -2 Po 3 Po -2 P* 4 Po -2 P* 3 P+ -2 P+ 4 P* -2 P* 4 f* -2 f, 5 p* -2 f* 5 f* -2 p* 6 P* -2 P i 3 s -2P* 3 do -2 p* 4 s -2p* 4 fo -2 p* 5 Po -2 P i 5 Po -2 P+ 6 -2 p i

5.1 1 0.92 3.07 3.29 4.22 4.5 1 4.96 5.14 5.36 0.65 2.65 3.55 4.07 4.17 4.77 5.22

5.07 0.93 3.09 3.29 4.22 4.51 4.95 5.15 5.32

5.11 0.92 3.10 3.28 4.21 4.49 4.94 5.14 5.32

5.12 0.91 3.07 3.29 4.21 4.46 4.92 5.31

the quantities that can be expanded to second order in k are the elements hij(k)of a 3 x 3 matrix (6 x 6 if spin orbit is included (Kane, 1956, 1957; Luttinger and Kohn, 1955; Luttinger, 1956)) whose eigenvalues are the E,@). Considering the a,@) as the components of a 3-component column vector ( n = 1,2,3) expressed on the basis of the eigenvectors of hij(k),it is advantageous to rewrite Eq. (1-33) using the natural basis states of hij(k).This leads to z h , ( k ) a j ( k ) + C V ( k ’ - k ) a i ( k )= E a j ( k ) k’

j

( 1 -34)

Now one can define slowly vaying functions F;.( r ) by the Fourier transformation Eq. (1-27) from ai ( k ) with k = 0 and get the generalization of Eq. (1-28):

2.64 4.08 4.17 4.76 5.52

hij( - i V, ) ~( r j)+ ~ ( r ) ~ ; =. (~r 4) j

4.70

Bloch states q),,k(r) (with n = 1, 2, 3) belonging to each of the three energy branches E,@). The first part of the derivation proceeds as for a single band and the generalization of Eq. (1 -26) becomes:

E, ( k )a, ( k )+ 2 V(k’- k )u, ( k ’ )= Ea, ( k ) k’ (1-33) This equation is diagonal in n and is apparently a simple to solve as for the single band extremum. However, the difficulty is to proceed further and transform it to a real space equation as in Eq. (1-28). The reason is that one can no longer define the derivatives of E,(k) near the valence band maximum at k = 0. This will be shown in detail in Section 1.3.2, following Kane’s derivation (Kane, 1956, 1957) in which it is shown that

(r)

(1-35)

As the h, are of second ord in k , this represents a set of coupled second order differential equations whose solution will lead to the expression of the envelope functions. Finally, in this approximation, the total wave function q (r)becomes, by using the basis set corresponding to h,:

q ( r > = Caj(k)eik.‘ u j k ( r )

(1 -36)

jk

which, if only k = 0 is involved, becomes (1 -37)

We do not discuss here the application of the method to hydrogenic acceptor states (details can be found in Bassani et al., 1974; Pantelides, 1978). We shall later see its use in quantum wells and superlattices. In such cases, the application of the envelope function approximation is complicated by the problem of boundary conditions which we discuss in Sec. 1.7.

14


1.2 The Calculation of Crystalline Band Structures We have seen that, for crystalline solids, the use of Bloch’s theorem allows us to demonstrate quite generally the existence of energy bands. However, in its derivation we have made the implicit assumption that one could write a Schrodinger equation for each electron taken separately. Of course this is in principle not permissible in view of the existence of electron-electron interactions and one should consider the N electron system as a whole. It has been shown that one can generalize the Bloch theorem to the one particle excitations of crystalline manyelectron systems. However, when this is done there is no exact method available to calculate these excitations (which correspond to the energy bands) in practice. One is left with approximate methods which are all based on the reduction of the problem to a set of separate one particle equations whose eigenvalues are used to compare the experimental one particle excitations. First, we thus give a brief account of most one-electron theories that have been used so far: Hartree, Hartree-Fock, local density, etc. We also discuss recent advances which have allowed to considerably improve the local density results (the so-called “G-W approximation”, which consists in a first order expansion of the electron self energy in terms of the screened electron-electron interaction). Usually, it is not necessary to include core electrons in calculations involving properties of the valence electrons. To achieve this separation one replaces the true atomic potentials by “first principles pseudopotentials” of which we give a short description. All this completely defines the single particle equations and, in Sec. 1.2.2 we present some techniques that can be used for their resolution. These techniques often involve a substantial amount of computa-

tion but they can be applied to simple crystals such as the zinc-blende semiconductors. However, there is a need for simpler empirical methods, either for physical understanding or as simulation tools for more complex systems. We describe two such methods in Sec. 1.2.3: the empirical pseudopotential method (EPM) and the empirical tight binding approximation (TBA).

1.2.1 Ab Initio Theories We describe here some basic methods that lead to approximate single particle equations. We begin with the Hartree approximation which is the simplest to derive and illustrates the general principles that are applied. We then discuss the Hartree-Fock approximation, local density theory, and its recent improvements via the G-W approximation.

1.2.1.1 The Hartree Approximation The full N electron Hamiltonian (for fixed nuclei) can be written: (1 -38)

where the hi are independent individual Hamiltonians containing the kinetic energy operator of electron i as well as its attraction by the nuclei. The second term in Eq. ( 1-38) represents the electron-electron interactions, r,, being the distance between electrons i and j . If one could neglect these, the problem would be exactly separable, i.e., one could obtain the solution of the full problem by simply solving the individual Schrodinger equations,

hi

(Pn,(ri)

= E,,

(Pn,(fj)r

(1 -39)

the full wave function being a product of individual wave functions (if one forgets for the moment the fact that it must be anti-

15

1.2 The Calculation of Crystalline Band Structures

symmetric) and the total energy being the sum of individual energies. The inclusion of the electron-electron interactions prevents the problem from being separable. However, one can find approximate individual equations by using a trial wave function (in the variational sense) which is of a separated form, i.e., it is a simple product of individual functions. For N electrons this gives

tion has been used in understanding the basic physics of atoms. It is not refined enough to be used for actual band structure calculations.

(1-40) n,

The unknown wave functions can be obtained by using the variational method, Le., by minimizing the average value of H with respect to the qn,.This leads to a set of individual equations which are the Hartree equations. However, these can be obtained directly by the following simple physical argument. If the problem can be separated, then the Hamiltonian of electron i will consist of the sum of its kinetic energy operator, its potential energy in the field of the nuclei, and its potential energy of repulsion with other electrons. This leads to the Schrodinger equation:

where the second term in the Hamiltonian represents the average electrostatic repulsion exerted on electron i by all the other electrons. There are as many equations (1-41) as there electrons, Le., N . Each oneelectron Hamiltonian contains the wave functions of the other electrons which are unknown. One has thus to proceed by iterations until a self-consistent solution is found, i.e., the wave functions injected into the hamiltonian are the same as the solutions of Eq. (1-41). The Hartree approxima-

- I --

Vn,(r1)~ n (ri , ) * . q n N (r1) Vn,(r2) (Pnl (r2) * * q n N (r21

JrVl-.

...

q n , ( r ~q n)2 ( T N ) . q n N ( r1 ~ * *

= E ~ n (ri ,1

(1 -43)

The result is the Hartree contribution plus a correction factor, the exchange term, due to antisymmetry in the electron permutations. It is important to notice that the q,,,(rj)are “spin orbitals”, i.e., products of a spatial part multiplied by a spin function (Slater, 1960). This is fundamental in order to maintain the Pauli principle.

16


The Hartree-Fock method is not very easy to apply numerically in view of the complexity of the exchange terms. Its application to covalent solids like diamond or silicon (Euwema et al., 1973; Mauger, Lannoo, 1977) leads to the overall correct shape of the energy bands, but a large overestimation of the forbidden gap. For instance, one gets 12 eV and 6 eV for diamond and silicon respectively compared to the experimental values of 5.4 eV and 1.1 eV respectively. Improvements on the Hartree-Fock approximation can be made including what are called correlation effects.

1.2.1.3 The Local Density Approximation The local density approximation is an extension of the Thomas-Fermi approximation based on the Hohenberg and Kohn theorem (Hohenberg and Kohn, 1964) which shows that the ground state properties of an electron system are entirely determined by the knowledge of its electron density e(r).The total energy of the interacting electron system can be written: ( 1-44)

+ j vex, ( r )e ( r m + Ex,@) where T represents the kinetic energy, the second term gives the electrostatic interelectronic repulsion, Vex, is the potential due to the nuclei, and E,, is the exchange correlation energy. A variational solution of the problem (Kohn and Sham, 1965) allows us to derive a set of one particle Schrodinger equations of the form: r

7

with t representing the one electron kinetic energy and ( 1 -46) occupied k

The practical resolution of Eqs. (1-45, 1-46) is, as we will discuss later, usually performed by replacing the atomic potentials by pseudopotentials, avoiding the explicit consideration of atomic core states. Originally these pseudopotentials were treated empirically, but now methods exist which allow us to determine them quantitatively from the properties of the free atoms (Hamann et al., 1979).The knowledge of these pseudopotentials plus the local density treatment allows a complete determination of the solutions yjk of Eq. (1 -45).The wave functions yjk can be calculated either using an expansion in plane waves or in orbitals localized on the atoms. Equations (1-45) and (1 -46) have to be solved in a self-consistent way. Up to now this formulation has been exact, the problem is that the quantity Ex,, and thus Vxc,is not known in general. The local density approximation then is based on the assumption that, locally, the relation between E,, and Q ( r ) is the same as for a free electron gas of identical density, which is known quite accurately. This approximation turns out to give satisfactory results regarding the prediction of the structural properties of molecules and solids. For instance, in solids (either with sp bonds or d bonds, as in transition metals) the cohesive energy, the interatomic distance, and the elastic properties are predicted with a precision better than 5 % in general. This remains true for diatomic molecules, except that the binding energy is overestimated by about 0.5 to 1 eV (see Cohen, 1983; Schluter, 1983 for recent reviews on the subject).

17


It is tempting to use the differences between the eigenvalues of Eq. (1-45) as particle excitation energies. This is not justified in general as seen by the predicted values for the energy gap E ~ = E ~ - E ” in semiconductors and insulators, taken as the difference in the energies of the first empty state and of the last filled state. The local density value of gG is always found to be substantially smaller than the experimental (Hamann, 1979). It is equal to 0.6 eV for silicon instead of 1.2 eV and even vanishes for germanium. The origin of the errors cannot be traced to the use of an overly simplified exchange and correlation potential such as LDA. This has been clearly shown by the almost identical results obtained using an improved exchange-correlation potential. The exchange-correlation term thus cannot be reduced to a simple local potential. 1.2.1.4 Beyond Local Density (the G-W Approximation)

To correct for the deficiencies of local density in defect calculations, one simple approach has been to use a “scissors” operator (Baraff and Schluter, 1984) which corrects for the band gap error by using a rigid shift of the conduction band states. It was later shown (Sham and Schluter, 1985, 1986; Perdew and Levy, 1983; Lannoo and Schliiter, 1985) that this procedure was closer to reality than expected, since the one electron exchange correlation potential of the local density formalism must experience a discontinuity across the gap. This is correctly handled by the scissors operator for bulk semiconductors but the applicability of the scissors operator to the defect levels is still questionable. In particular, in the case of extended defects such as surfaces, it is doubtful that the defect states are correctly obtained. The advantage of this correc-

tion is mainly that it does not add any computational requirements when compared to conventional LDA calculations. A more sophisticated way of improving the density functional theories is to evaluate the electron self-energy operator Z(r,r ’, E ) (Lannoo and Schluter, 1985; Hybertsen and Louie, 1985; Godby, 1986). This Zcontains the effects of exchange and correlation. It is non-local, energy-dependent, and non-hermitian. Its non-hermiticity means that the eigenvalues of the new one particle Schrodinger equation, ( t + vest+ VH) q,&) +

(1 -47)

I

+ dr’a r , r’,En,) W,,(r‘) = E,, qn,(r) will generally be complex. The imaginary part gives the lifetime of the quasiparticle, and V , is the Hartree-potential. The self-energy operator can be estimated using the G-W approximation (Hedin and Lundquist, 1969). The self-energy is expanded in a perturbation series of the screened Coulomb interaction, W The first term of the expansion corresponds to the Hartree-Fock approximation. Details can be found in Lannoo and Schliiter (1985); Hybertsen and Louie (1985); Godby et a]. (1986, 1987); Hedin and Lundquist (1969). These early works have been completed by more recent calculations (Rohlfing et al., 1993; Northrup et al., 1991; Northrup, 1993; Blase et a]., 1994). The results show a great improvement on the values predicted for the gap of semiconductors as evidenced in Louie (1996). 1.2.1.5 The Pseudopotential Method

The full atomic potentials produce strong divergences at the atomic sites in a solid. These divergences are related to the fact that these potentials must produce the atomic core states as well as the valence states.

18


However, the core states are likely to be quite similar to what they are in the free atom. Thus the use of the full atomic potentials in a band calculation is likely to lead to unnecessary computational complexity since the basis states will have to be chosen in such a way that they describe localized states and extended states at the same time. Therefore it is of much interest to devise a method which allows us to eliminiate the core states, focusing only on the valence states of interest which are easier to describe. This is the basis of the pseudopotential theory. The pseudopotential concept started with the orthogonalized plane wave theory (Cohen et al., 1970). Writing the crystal Schrodinger equation for the valence states,

( T + v,/ly)=Elv)

(1 -48)

one has to recognize that the eigenstate 1 y ) is automatically orthogonal to the core states 1 c ) produced by the same potential V. This means that 1 v ) will be strongly oscillating in the neighbourhood of each atomic core, which prevents its expansion in terms of smoothly varying functions, like plane waves, for instance. It is thus interesting to perform the transformation

lV)=(] - P )

lv)

( 1-49)

where P is the projector onto the core states ( 1-50)

1 q )is thus automatically

orthogonal to the core states and the new unknown lq) does not have to satisfy the orthogonality requirement. The equation for the “pseudostate“ 19) is:

( T + V)Cl - P ) l v ) = E ( l - P ) l ~ p ) (1-51) Because the core states I C ) are eigenstates of the Hamiltonian T + V with energy, E,,

one can rewrite Eq. ( -51) in the form

{ T +V+C(E-E,) C

4 (CI}

Icp) = E

Iv) (1-52)

The pseudo-wave function is then the solution of a Schrodinger equation with the This new same energy eigenvalue as 1). equation is obtained by replacing the potential V by a pseudopotential (1 -53)

This is a complex non-local operator. Furthermore, it is not unique since one can add any linear combination of core states to I cp) in Eq. (1-52) without changing its eigenvalues. There is a corresponding non-uniqueness in Vps since the modified 1 cp) will obey a new equation with another pseudopotential. This non-uniqueness in Vps is an interesting factor since it can then be optimized to provide the smoothest possible Icp), allowing rapid convergence of plane wave expansions for 19).This will be used directly in the empirical pseudopotential method. Recently, so-called “first principles” pseudopotentials have been derived for use in quantitative calculations (Hamann et al., 1979). First of all, they are ion pseudopotentials and not total pseudopotentials as those discussed above. They are deduced from free atom calculations and have the following desirable properties: (1) real and pseudovalence eigenvalues agree for a chosen prototype atomic configuration, (2) real and pseudo-atomic wave functions agree beyond a chosen core radius r,, (3) total integrated charges at a distance r > r, which agree (norm conservation), and (4) logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives which agree for r > r,. These properties are crucial for the pseudopotential to have optimum transferability among a variety of chemical environments, allowing self-con-

19


sistent calculations of a meaningful pseudocharge density.

1.2.2 Computational Techniques We now will discuss some techniques which allow us to calculate energy bands in practice. They are simply based on an expansion of the eigenfunction of the one-electron Schrodinger equation in some suitable basis functions, plane waves, or localized atomic orbitals. These seem to be, by far, the most commonly adopted methods at this time.

obtained by mixing the plane wave e* only with other plane waves whose wave vector is k + G,where G is any reciprocal lattice vector, and not with all plane waves with arbitrary wave vectors. Of course, we are only interested in low lying states so that we can truncate the expansion of Eq. (1 -54) at some maximum value of [GI, which we label G,,,. The one particle Hamiltonian is thus expressed on this basis as a finite matrix of elements: '

1.2.2.1 Plane Wave Expansion Plane waves form a particularly interesting basis set for crystalline band structure calculations in conjunction with the use of pseudopotentials. As we shall see below the total pseudopotential can be expressed as a sum of atomic contributions. These consists of a bare ionic part which will be screened by the valence electrons. The resulting atomic pseudopotentials are often assumed to be local, i.e. to be simple functions of the electron position. However, in general, they should be operators having a non-local nature, as discussed later. The best starting point is the expression Eq. (1-9) of the Bloch function in which one makes use of the fact that the function uk(r) is periodic and can thus be expanded as a Fourier series: (1-54a) G

where G are reciprocal lattice vectors. This leads to the natural plane wave expansion for the wave function:

uk ( G )ei(' + C ) ' r

rpk ( r )=

(1-54 b)

G

The gain due to the Bloch theorem is that any Bloch state &(r) with wave vector k belonging to the first Brillouin zone is

which readily becomes

(1 - 5 5 )

A2

Hc,c~(.k)=-~k+G12SG,G'+vG,G4k) 2m (1 -56) where the second term is the potential matrix element. In the case where Vis a simple function of r , this matrix element can be reduced to: VG,Gt(k)= V ( G ' - G )

(1-57)

Such plane wave expansions can be used in different contexts. We will later develop an application known as the empirical pseudopotential method (EMP). One can also apply these expansions to first principles calculations. This is formally easy in the local density context where the potential takes a simple form. It becomes more complex, but still can be adapted, in the HartreeFock theory or even in the G-W approximation as used in Hybertsen and Louie (1985, 1986) and Godby et al. (1986).

1.2.2.2 Localized Orbital Expansion In this expansion the wave function is written as a combination of localized orbitals centered on each atom: (1-58) i,a

20


where via is the ath free atom orbital of atom i, at position R j . As each complete set of such orbitals belonging to any given atom forms a basis for Hilbert space, the whole set of cp,, is complete, i.e., the CP,, are no longer independent and Eq. (1-58) can yield the exact wave function of the whole system. In practice one has to truncate the sum over a in this expansion. In many simplified calculations it has been assumed that the valence states of the system can be described in terms of a “minimal basis set” which only includes free atom states up to the outer shell of the free atom (e.g., 2s and 2p in diamond). It is that description which provides the most appealing physical picture, allowing us to clearly understand the formation of bands from the atomic limit. The “minimal basis set” approximation is also used in most semi-empirical calculations. When the sum over a in Eq. (1 -58) is limited to a finite number, the energy levels E of the whole system are given by the secular equation: det ( H - E S 1 = 0

( 1-59)

where H is the Hamiltonian matrix in the atomic basis and S the overlap matrix of elements: Sia, j p

= ( P i a I Vjp)

(1 -60)

These matrix elements can be readily calculated, especially in the local density theory, and when making use of Gaussian atomic orbitals. The problem, as in the plane wave expansion, is to determine the number of basis states required for good numerical accuracy. An interesting discussion on the validity of the use of a minimal basis set has been given by Louie (1980). Starting from the minimal basis set 1 cp,,) one can increase the size of the basis set by adding other atomic states Ix,,), called the peripheral states,

which must lead to an improvement in the description of the energy levels and wave functions. However, this will rapidly lead to problems related to overcompleteness, Le., the overlap of different atomic states will become more and more important. To overcome this difficulty, Louie proposes three steps to justify the use of a minimum basis set. These are the following: Symmetrically orthogonalize the states I qia)belonging to the minimal basis set between themselves. This leads to an orthogonal set I gia). The peripheral states /xi,) overlap strongly with the It is thus necessary to orthogonalize them to these I pia), which yield new states lijii,) defined as: I i i i i , > = Ixip)-C

Ja

lea)

(ealXiii,>

(1-61)

Ixi,)

3. The new states are then orthogonalized between themselves leading to a new set of states Ifiii,). Louie has shown that, at least for silicon, the average energies of these atomic states behave in such a way that, after step 3, the peripheral states / X j p ) are much higher in energy and their coupling to the minimal set is reduced. They only have a small (although not negligible) influence, justifying the use of the minimal set as the essential step in the calcuation. The quantitative value of LCAO (linear combination of atomic orbitals) techniques for covalent systems such as diamond and silicon was first demonstrated by Chaney et al. (1 97 1 ). They have shown that the minimal basis set gives good results for the valence bands and slightly poorer (but still meaningul) results for the lower conduction bands. Such conclusions have been confirmed by Kane (1976), Chadi (1977), and Louie (1980) who worked with pseudopotentials instead of true atomic potentials.


The great asset of the minimal basis set LCAO calculations is that they provide a direct connection between the valence states of the system and the free atom states. This becomes still more apparent with the TBA (tight binding approximation) which we shall later discuss and which allows us to obtain extremely simple, physically sound descriptions of many systems.

1.2.3 Empirical Methods Up to very recently, first principles theories, sophisticated as they may be, could not accurately predict the band structure of semiconductors. Most of the understanding of these materials was obtained from less accurate descriptions. Among these, empirical theories have played (and still play) a very important role since they allow us to simulate the true energy bands in terms of a restricted number of adjustable parameters. There are essentially two distinct methods of achieving this goal: the tight binding approximation (TBA) and the empirical pseudopotential method (EPM).

1.2.3.1 The Tight Binding Approximation This can be understood as an approximate version of the LCAO theory. It is generally defined as the use of a minimal atomic basis set neglecting interatomic overlaps, i.e., the overlap matrix defined in Eq. (1-60) is equal to the unit matrix. The secular equation thus becomes det IH-EZI = O

( 1-62)

where 1 is the unit matrix. The resolution of the problem then requires the knowledge of the Hamiltonian matrix elements. In the empirical tight binding approximation these are obtained from a fit to the bulk band structure. For this, one always truncates the

21

Hamiltonian matrix in real space, i.e., one only includes interatomic terms up to first, second, or, at most, third nearest neighbors. Also, in most cases one makes use of a twocenter approximation as discussed by Slater and Koster (1954). In such a case, all Hamiltonian matrix elements (qiaI H I qjp) can be reduced to a limited number of independent terms which we can call Hap(i,j) for the pair of atoms ( i , j ) and the orbitals (a,P).On an “s, p” basis, valid for group IV, 111-V, and 11-VI semiconductors, symmetry consideration applied to the two-center approximation only give the following independent terms:

H,, (i,j)?H,,

(id

(1 -63)

where H, is strictly zero in a two-center approximation and s stands for the s orbital, CT the p orbital along axis i, j with the positive lobe in the direction of the neighboring atom, and n a p orbital perpendicular to the axis i, j . With these conventions, all matrix elements are generally negative. Similar considerations apply to transition metals with s, p, and d orbitals. Simple rules obtained for the H a p ( i , j ) in a nearest neighbor’s approximation are given in Harrison (1980). They are based on the use of free atom energies for the diagonal elements of the tight binding hamiltonian. On the other hand, the nearest neighbor’s interactions are taken to scale like d-* (where d is the interatomic distance) as determined from the free electron picture of these materials which will be discussed later. For s, p systems, this gives

where d is expressed in A.

Next Page

22


Such parameters nicely reproduce the valence bands of zinc-blende semiconductors but poorly describe the band gap and badly describe the conduction bands. Improvements on this description have been attempted by going to the second nearest neighbors (Talwar and Ting, 1982) or by keeping the nearest neighbors treatment as it is but adding one s orbital (labelled s*) to the minimal basis set (Vogl and Hjalmarson, 1983). The role of this latter orbital is to simulate the effect of higher energy d orbitals which have been shown to be essential for a correct simulation of the conduction band. The quality of such a fit can be judged from Fig. 1-6, which shows that the lowest conduction bands are reproduced much more correctly. Fairly recently it has also been shown that the replacement of s* by true d orbitals improves the simulation in a striking manner (Priester).

1.2.3.2 The Empirical Pseudopotential Method One advantage of the tight binding approximation is that it provides a natural way of relating the electronic properties of a solid to the atomic structure of its constituent atoms. As will be discussed later, it is also the most appropriate way to calculate the properties of disordered systems. However, for crystalline semiconductors the empirical pseudopotential method seems to be the most efficient way to get a good overall description of both the valence and the conduction bands. The basis of this method is the plane wave expansion of the wave function given by Eq. ( 1-54) plus the use of a smooth pseudopotential. The matrix to be diagonalized was derived in Eq. (1-56) but now we pay more attention to the potential matrix elements. In EPM one assumes that the selfconsistent crystal pseudopotential can be written as a sum of atomic contributions, i.e.,

va(r - Rj - r a )

V ( r )=

( 1-65)

J3a

where j runs over the unit cells positioned at Rj and a is the atom index, the atomic position whithin the unit cell being given by ra. Let us first assume that the v, are ordinary functions of r, or, in other words, that we are dealing with local pseudopotentials. In that case the matrix elements [Eq. (1 -57)] of V between plane waves become: ( k + ~ l ~ l k + ~ ’ ) = -15 1 W

I

I

L

r

X U,K

(1-66)

r

Figure 1-6. Comparison of the sp3 s* description of silicon ( - - - ) with a more sophisticated calculation (-). The vertical axis represents energies in eV, the horizontal axis the wave vector along symmetry axes in the Brillouin zone. Note that the valence band is practically perfectly reproduced.

where Q is the volume of the unit cell. Suppose that there can be identical atoms in the unit cell. Then the sum over a can be expressed as a sum over groups /3 of identical atoms with position specified by a second

Previous Page

1.3 Comparison with Experiments for Zinc-Blende Materials

index y (i.e. ra = ryp).Calling n the number of atoms in the unit cell we can write

= E S’(G’-G)vp

(G’-G) (1-67)

P where Sp(G) and vp ( G ) are, respectively, the structure and form factors of the corresponding atomic species, defined by

and

(1 -68)

n vp(G)=- ,fvP(r)eiG“d3r $2 In practice, the empirical pseudopotential method treats the form factors vp (G) as disposable parameters. In the case where the vp (r)are smooth potentials their transforms vp(G) will rapidly decay as a function of IG 1 so that it may be a good approximation to truncate them at a maximum value of G,. For instance, the band structure of tetrahedral covalent semiconductors like Si can be fairly well reproduced using only the three lower Fourier components v(lG1) of the atomic pseudopotential. There are thus two cut-off values for (G1 to be used in practice: , limits the number of plane waves one, G and thus the size of the Hamiltonian matrix; the other one, G,, limits the number of Fourier components of the form factors. We shall later give some practical examples. The use of a local pseudopotential is not fully justified since, from Eq. (1.52), it involves, in principle, projection operators. It can be approximately justified for systems with s and p electrons. However, when d states become important, e.g., in the conduction band of semiconductors, it is necessary to use an operator form with a projection operator on the 1 = 2 angular components.

23

1.3 Comparisonwith Experiments for Zinc-Blende Materials In this section we apply methods which allow us to understand the general features of the band structure of zinc-blende materials. We detail, to some extent, simple models based on tight binding or the empirical pseudopotential. We also discuss briefly the results of the most sophisiticated recent calculations. We then concentrate on the general treatment of the band structure near the top of the valence band, using the k-p perturbation theory as in the work by Kane (1 956; 1957). In the third part, we examine the optical properties of these materials putting particular emphasis on excitonic states. Finally, we give a comparison of predicted band structures with photo-emission and inverse photo-emission data, in the light of what was done on the basis of non-local empirical pseudopotentials.

1.3.1 The General Shape of the Bands In this section we start from two points of view: (i) the molecular or bond orbital model derived from tight binding and (ii) the nearly free electron picture. We show that both give rise to similar qualitative results at least for the valence bands.

1.3.1.1 The Tight Binding Point of View Consider an A-B compound in the zinc blende structure where the atoms have tetrahedral coordination. The minimal atomic basis for such systems consists of one s and three p orbitals on each atom. One could solve the Hamiltonian matrix in this basis set and get the desired band structure directly. However, one can get much more insight into the physics by performing a basis change such that, in the new basis set, some matrix elements of the Hamiltonian

24


will be much larger than the others. This will allow us to proceed by steps, treating first the dominant elements and then looking at the corrections due to the others. This is the general basis of molecular (Harrison, 1973; Lannoo and Decarpigny, 1973) or bond orbital models, two names for the same description. The natural basis change is to build sp3 hybrids of the form

Of course these states will be strongly degenerate since their degeneracy is equal to the number N of bonds in the system. The wave function of the bonding and antibonding states will take the form, for a bond ij connecting two neighbors i and j ,

Wa,ij

=

( 1-69)

where qs,is the s orbital of atom i and qp,l, is one of its p orbitals pointing from i to one of its nearest neighbors j . By doing this for each atom, each bond in the system will be characterized by a pair of strongly overlapping hybrids q,, and qJ, as shown i n Fig. 1-7. It is clear that the dominant interatomic matrix elements of the Hamiltonian are given by ( 1-70) (VI, I H I % I ) = - P B > 0 are while the diagonal elements ( qiJI H I qlJ) equal to the average sp3 energies of the atoms, which we denote E, for atom A and E , for atom B. In the first step we neglect all other matrix elements. The problem is then equivalent to a set of identical diatomic molecules, each one leading to one bonding and one antibonding state of energy:

k q Y. .- qJ ..1 Jl

(1 -72)

+ y2

In the following discussion we take the convention that i is an A atom a n d j a B atom. As there are two electrons per bond, the ground state of the system corresponds to completely filled bonding states and empty antibonding states. This description defines the “molecular” or “bond orbital model” in which the bonding states give a rough account of the valence band and the antibonding states, of the conduction band. This model has been extremely successful in describing semiquantitatively the trends in several physical properties of these materials: ionicity, effective charges, dielectric susceptibilities, average optical gaps, and even cohesive properties (see Harrison (1 980) for more details). With such a simple starting point, the formation of the band structure is easy to describe. The inclusion of further interactions which were neglected in the molecular model will tend to lift the degeneracy of the bonding and antibonding states. Exactly the same arguments as those developed in Sec. 1.1 lead to the conclusion that there will be the formation of a bonding band from the bonding states. For this we write the wave function Vb as a combination of all

vb,v:

Figure 1-7. Pair of sp3 hybrids involved in one bond, as defined i n Eq. (1-69).

Wb

=

ab,ij Wb,ij pairs ij

(1 -73)

25


The leading correction term in the Hamiltonian matrix will be the interaction between two adjacent bonds which takes one of two values, AA or AB, depending on the common atom. Projecting Schrodinger’s equation on these states one gets the set of

where the sums are over adjacent bonds having an A or B atom in common. At this stage it is interesting to introduce the following sums si

=

Ubij

and

sj

j

= E abij

AB sj ( 1 -76) Summing this either overj or i, one gets two equations: jei

sj

(1 -77)

(E-Eb+AA-3A,)Sj = A A C S ~ icj

where the sums are over the nearest neighbors of one given atom. Injecting the second Eq. (1 -77) into the first one gives with A = AA + A B and S =

2

2

[(E-Eb)2 - 4 A ( E - E b ) - 1 2 S 2 ) S i = = (A2 - S 2 )

C Si2

(1 -78)

‘2

where now the sum is over the second nearest neighbors of atom i. This set of equations on sublattice A is just the same as what would be obtained for a tight binding s band on an f.c.c. lattice. This leads to the following solutions:

E=Eb+ +2 4

(1 -79)

[4A’

2

2

2

2

a being the lattice parameter, and k,, k,, and k, the components of the wave vector along the cube axes. It is clear that ‘p varies continuously from 12 to -4. The extrema of the bands ( c p = 12) are then given by:

while for ‘p = - 4 a gap is opened in the band, its limits being given by

+ AA + AB) Ubjj = AA si

( E - Eb - 3 AA + AB)si = AB

( 1 -80)

2

(1 -75)

i

so that one can rewrite Eq. (1-74) as (E-

with

+ 12 6’ + (A2 - 6’) 411”’

E = Eb + 2 4

f4

161

(1 -82)

Equation (1-79) gives only two bands while we started with four bonding orbitals per atom of the A sublattice. The two bands which appear to be missing can be obtained by noting that Eq. (1 -76) has a trivial solution for which all the Siare zero, with nonzero abjj if

E = Eb - 2 4

(1-83)

This is the equation of a twofold degenerate flat band with pure p character, since all Si and Sj are zero. To summarize the results, one can say that the interaction between bonding orbitals has broadened the bonding level into a valence band consisting, in the present model, of two broad bands and a twofold degenerate flat band. Exactly the same treatment can be applied to the antibonding states by changing Eb into E, and defining A, and AB describing the interactions between antibonding states. This antibonding band will thus represent the conduction band. The resulting band structure is compared in Fig. 1-8 to a more sophisticated calculation from Chaney et al. (1971) (its

26


m---

The advantage of the simple picture we have just detailed is that it can be generalized, as we shall see, to a lot of other situations such as covalent systems with lower coordination (Sec. 1.4) or to non-crystalline and amorphous semiconductors. Furthermore, all tight binding descriptions with parametrized interactions give results which are in good correspondence with those we have derived, with the minor corrections we have mentioned. One such empirical model is the sp3 s* description of Vogl et al. (1 983) which leads, for GaAs, to the band structure of Fig. 1-9, which can be shown to be in good agreement with experiments for the valence band and the lowest conduction band.

1.3.1.2 The Empirical Pseudopotential Method

r1 Figure 1-8. Comparison of the simple model description of diamond (- - - ) with a more sophisticated calculation (-). The vertical axis corresponds to energies in eV.

parameters have been adjusted to give overall agreement). It can be seen that it already reproduces the essential features of the valence band. The inclusion of further matrix elements left out from the previous simple picture will have the following qualitative effects: i) the bonding-antibonding interactions lead to a slight repulsion between the valence and conduction bands, and ii) the inclusion of interactions between more distant bonds induces some dispersion into the flat bands.

A major improvement in the detailed description of the bands of tetrahedral semiconductors has been achieved with the use of empirical pseudopotentials. Let us then first discuss its application to purely covalent materials like silicon and germanium. The basis vectors of the direct zinc-blende lattice are a/2 (1 lo), a/2 (01 l), and a/2 (101). The corresponding basis vectors of the reciprocal lattice are 2 n l a (1 1 I ) , 2 d a (11l ) , and 2 x / a (151). The reciprocal lattice vectors G which have the lowest square modulus are the following, in increasing order of magnitude:

000 111

200 220 31 1

0 3 4 8 11

(1 -84)

For elemental materials like Si and Ge there is only one form factor v(G)but we


27

Figure 1-9. Comparison between the sp3 s* band and structure of GaAs (-) the empirical pseudopotential one (- - -). Vertical scale: energies in eV, horizontal scale: k values.

have seen in Eq. (1.67) that the matrix element of the potential involves a structure factor which is given here by: S ( G ) = COS G . Z

(1 -85)

where the origin of the unit cell has been taken at the center of a bond in the (1 11) direction and where Zis thus the vector d 8 (1 11). For local pseudopotentials this matrix element ( k + GI VI k + G') can be written V ( G )and is thus given by: V ( G )= v ( G ) COS ( G -

Z)

(1-86)

The structure factor part is of importance since, among the lowest values of IGl quoted in Eq. (1-84), it gives zero for 2 x l a (2, 0,O). If one indexes V ( G )by the value tak) ~ then only the en by the quantity ( U / ~ T GG2, values V,, V,, and V , , are different from zero. It has been shown (in Cohen and Bergstresser, 1966) that the inclusion of these three parameters alone allows us to obtain a satisfactory description of the band struc-

ture of Si and Ge. This can be understood simply by the consideration of the free electron band structure of these materials which is obtained by neglecting the potential in the matrix elements Eq. (1-56) of the Hamiltonian between plane waves. The eigenvalues are thus the free electron energies A2/2m Ik + G l 2 which, in the f.c.c. lattice, lead to the energy bands plotted in Fig. 1-10. The similarity is striking, showing that the free electron band structure provides a meaningful starting point. The formation of gaps in this band structure can be easily understood at least in situations where only two free electron branches cross. To the lowest order in perturbation theory, one will have to solve the 2 x 2 matrix.

;:1

-1k

+GI' V ( G ' - G )

V * ( G ' - G ) -Ik+G'/' 2m

1

(1-87)

28


12 8 4

2

-

z

P 0

:

-4

2

U

-8 -12

L

A

r

A

z

XK,U

r

L

r

X

K,U

r

Free-electron bands

True bands

Figure 1-10. Correspondence between free-electron and empirical pseudopotential bands, showing how degeneracies are lifted by the pseudopotential.

The esulting eigenvalues are

E(k

+IV(G’-G)I

7 112

}

( 1 -88)

whose behavior as a function of k is pictured in Fig. 1-1 1. The conclusion is that there

--

PV(G’-G)

is the formation of a gap at the crossing point whose value is 21 V(G’- G)I. Note that for this to occur the crossing point at G’+G k=-must lie within the first Bril2 louin zone or at its boundaries. For points where several branches cross, one will have a higher order matrix to diagonalize but this will generally also result in the formation of gaps. This explains the differences between the free electron band structure and the actual one in Fig. 1-10. The number of parameters required for fitting the band structures of compounds is different in view of the fact that there are now two different atoms in the unit cell with form factors v,(G) and uB(G).The matrix elements V ( G )of the total pseudopotential will thus be expressed as ( 1 -89) V(G)=VS(G)cos (G.z)+iVA(G)sin(G.z)

k

Figure 1-11. Opening of a gap in the nearly free electron method. The full line corresponds to the two free electron branches, the dashed lines to the two branches split by the potential Fourier component.

where vSand vAare equal to (v,+ vB)/2and (v,-u,)/2, respectively. The number of fitting parameters is then multiplied by 2, the symmetric components Vi, Vi, and Vf, being close to those of the covalent materials and the antisymmetric components be-


ing V t , Vf, and V& since the antisymmetric part of V , vanishes.

1.3.2 The k-p Description and Effective Masses We have already seen for hydrogenic impurity states (Sec. 1.1) that the concept of effective masses near a band extremum is very powerful. This will prove still more important for heterostructures which we discuss later. In any case it is desirable to provide a general framework in which to analyze this problem. This is obtained directly via the k-p method which we present in this section. The basis of the method is to take advantage of the crystalline structure which allows us to express the eigenfunctions as Bloch functions and to write a Schrodingerlike equation for its periodic part. We start from (1-90) + v]eik uk(r)= ~ ( kei)k .ruk( r)

{&

where we have written the wave function in Bloch form. We can rewrite this in the following form:

L m 1

( P + A k ) 2+ V u k ( r ) = E ( k ) u k ( r )(1-91)

which is totally equivalent to the first form. To solve this we can expand the unknown periodic part uk(r)on the basis of the corresponding solutions at given point ko, which we label unk0(r): cn ( k )un, ko ( r )

u k ( r )=

(1-92)

n

The corresponding solutions are the eigenvalues and eigenfunctions of the matrix with the general element ( 1 -93)

29

We now use the fact that u n , k o is an eigenfunction of Eq. (1.91) fork =k,, with energy En@,). This allows us to rewrite the matrix element Eq. (1-93) in the simpler form:

with Pnnt(kO)

(1-95)

= (un,ko IP lun’,ko)

Diagonalization of the matrix A ( k ) given by Eq. (1-94) can give the exact band structure (an example of this is given in Cardona and Pollak, 1966). However, the power of the method is that it represents the most natural starting point for a perturbation expansion. Let us illustrate this first for the particular case of a single non-degenerate extremum. We thus consider a given non-degenerate energy branch En(k) which has an extremum at k = k , and look at its values for k close to k,. The last term in Eq. (1 -94) can then be considered as a small perturbation and we determine the difference En ( k )E,@,) by second order perturbation theory applied to the matrix A ( k ) .This gives A* En(k)=En(ko)+-(k-ko)2 2m A2

c [(k

+2 m

n’#n

+

(1-96)

- ko ) * P n n f I[(k - ko 1* Pn’n 1

En (ko 1- Ent(ko )

which is the second order expansion near k , leading to the definition of the effective masses. The last term in Eq. (1 -96) is a tensor. Calling 0 a its principal axes, one gets the general expression for the effective masses m,*

30


This shows that when the situation practically reduces to two interacting bands, the upper one has positive effective masses while the opposite is true for the lower one. point for This is what happens at the GaAs, for instance. Another very important situation is the case of a degenerate extremum, i.e., the top of the valence band in zinc blende materials which occurs at k = 0. We still have to diagonalize the matrix Eq. (1 -94) taking k , = 0 and, fork = 0, the last term can still be treated by the second order perturbation theory. By letting i andj be two members of the degenerate set at k = 0 and 1, any other state distant in energy, we now must be apply the second order perturbation theory on a degenerate state. As shown in standard textbooks (Schiff, 1955) this leads to diagonalization of a matrix: r

-

7

The top of the valence band has threefold degeneracy and its basis states behave like atomic p states in cubic symmetry (Le., like the simple functions x , y, and I). The second order perturbation matrix is thus a 3 x 3 matrix built from the last term in Eq. (1-98) which, from symmetry, can be reduced to (Kane, 1956, 1957: Kittel and Mitchell, 1954: Dresselhaus et al., 1955)

Lkt

+ M(kZ + k : ) N k y kz

( P a ) ; /( P p ) / j i

E,(o)-E,(o)

L * S = 1/2 ( J 2 - L2 - S 2 )

Lk;

(1-100)

(1-101)

where J = L + S . Because here L = l and S = 1/2, J can take two values J = 3/2 and J = 112. From Eq. (1-101) the J = 3 / 2 states will lie at higher energy than the J = 1/2 ones and, if the spin orbit coupling constant is large enough, these states can be treated separately. The top of the valence band will then be described by the J = 3/2 states leading to a 4 x 4 matrix whose equivalent Hamiltonian has been shown by Luttinger and Kohn (1955) to be:

H = m~ { ( y , + ~ y2 2 ) ~ a- yk 22J ;Z-

where

N k x, kkz=

where L, M , and N are three real numbers, all of the form: A?

2m diagonal wich define the h&) matrix of Sec. 1.1.4 to be applied in effective mass theory to a degenerate state. Up to this point we have not included spin effects and in particular spinorbit coupling, which plays an important role in systems with heavier elements. If we add the spin variable, the degeneracy at the top of the valence band is double and the k . p matrix becomes a 6 x 6 matrix whose detailed form can be found in (Bassani et al., 1974; Altarelli, 1986; Bastard, 1988). One can slightly simplify its diagonalization when the spin orbit coupling becomes large, from the fact that

h‘ kx k,

[Lkf +M(k: +kz)

-E m’

A2 k 2 It is this matrix plus the term - on its

CY,

p = x , v or z.

1

(1 -99)

+Mik: +k:)]

Finally, as shown by Kane (1956, 1957), it can be interesting to treat the bottom of the conduction band and the top of the valence band at k = 0 as a quasi-degenerate


system, extending the above described method to a full 8 x 8 matrix which can be reduced to a 6 x 6 one if the spin orbit coupling is large enough to neglect the lower valence band.

which can be expressed as an integral over the surface S(Aw) in k space such that E,@) - E , ( k ) = ho.This gives

2 Jvc(w)=y S(0)

1.3.3 Optical Properties and Excitons One of the major sources of experimental information concerning the band structure of semiconductors is provided by optical experiments. In particular fine structures in the optical spectra might reflect characteristics of the band structure. To see this in more detail let us discuss the absorption of the light which is proportional to the imaginary part of the frequency dependent dielectric constant, i.e.,

dS

IV,(Ec(k)- EV(k))ls (1-106)

It is clear from this expression that important contributions will come from critical points where the denominator of Eq. (1- 106) vanishes. These can originate from k points where one has the separate conditions

VkEc(k)= 0, V k E , ( k ) = 0

(1-103)

where the optical matrix element is:

4, = ( V c , k , I e'%' vr I

V V d

(1-104)

(1-107)

Critical points of this kind only occur at high symmetry points of the Brillouin zone (such as k = 0, for instance). Other critical points are given by

Vk (E, ( k )- E" ( k ) )= 0 *6(Ec-E,-ho)d3k

31

(1-108)

The behavior near such Van Hove singularities can be discussed quite generally by expanding E, - E, to the second order in k , around the singular point ko. This formally gives 3

C

Ec-Ev=Eo+ ~ ~ ( k , - k o , ) ~(1-109) x being the wave vector of light. From the a=l Bloch theorem this matrix element is nonzero only if k' is equal to k + x + G(G is the and the qualitative behavior of the joint denreciprocal lattice vector). As I X I is much sities of states J,,(w) near such a point desmaller than the dimensions of the first Brilpends on the sign of the a,. Typical results louin zone, this means that the transition is are shown in Fig. 1-12, showing that, from vertical within the Brillouin zone @e., it the shape of the absorption spectrum, one occurs at fixed k ) . Fine structures in E ~ ( c L ) ) can infer a part of the characteristics in the due to allowed transitions can be studied by band structures. As an illustration, we show discarding the k dependence of the optical in Fig. 1- 13 the curves e2(o)for silicon and matrix element M,,(k). In a narrow energy germanium in comparison to the predicted be-0 ) curves (Greenaway and Harbeke, 1968). range around such a structure, ~ ~ ( comes proportional to the joint densities of Up to now we have only discussed optistates cal transitions between one particle states in which an electron is excited into the conL (1-105) J,,(w)=-. duction band, leaving a hole in the valence (2 3o3 band. This is permissible if we treat the . 6 ( Ec ( k ) - E, ( k) - A W ) d3k electron and hole as independent particles.

32


at hv = E,, the energy gap, can show lines m*e4 1

at Eg

(dl

\*

( c ~

w.

\ I I

vexc

I

3.5

E (eV)

is a suitable

~

~

a(k,, kh) @(ke,kh)

=

( l - l lo>

where the @correspond to the excited states obtained from the ground state by exciting a valence band electron of wave vector kh to a conduction band state k,. To use the effective mass approximation we introduce a two particle envelope function by the Fourier transform:

However, these quasi-particles have opposite charges and attract each other via an effective potential -e2/&r. This coulombic potential can give rise to localized gap states such as hydrogenic impurities, so that the absorption spectrum, instead of starting

2.5

~

ke,kh

w.

1.5

,, where m*

t n' i

effective mass. Again the justification of this proceeds via the effective mass theory, but in a way slightly more complex than for impurities. To find the excitonic wave function, we can write the total wave function of the excited states in the form (Knox, 1963; Kittel, 1963):

Figure 1-12. Schematic joint densities of states near the critical points for different situations (see text).

0

~

4.5

5.5

6.5

Figure 1-13. Experimental and predicted e2(w ) for Ge.

33


For simple band extrema with isotropic effective masses, it can be shown along the same lines as in Sec. 1.1 that F(re,rh)obeys the effective mass equation:

tions, Le., only if k, = kh = k . This matrix element is identical to the M v c ( k )defined above and we take it to be constant over the small range of k involved. We then get Mext = ~

v

c

a(k,k)

(1-1 16)

k

From the definition of the envelope function Eq. (1-111) we see that C a ( k , k ) One can separate the center of mass and relative motion in this Hamiltonian in such a way that the total energy becomes:

k

is equal to F(r,r)d3r where we take re-rh=r.We thus obtain the final result: IMexcl2

A2k2 - m*e4 1 E = Eg + 2~ ~ E n2~

(1-113) A

~

where M and m* are the total and reduced masses respectively and k is the wave vector for the center of mass motion. From this it is clear that the lowest excited states are those for k = 0 and these give rise to the hydrogenic lines. It is interesting to determine the oscillator strength for exciton absorption for comparison with one particle transitions. We have seen before that, at a given frequency, the strength of the absorption is determined by the optical matrix element. For many electron states, this element is given by Mexc

= (VOIC P i I V e x , )

(1-114)

i

= lMvc121 I F ( r , r )d3 r I*

(1-1 17)

For the simple model we have just considered, the lowest exciton wave function is

where Vis the volume of the specimen, R is the center of mass position and a the exciton Bohr radius, thus leading to 2 2 v IMexcl =/MvcI -

nu3

(1-119)

This is to be compared to the one particle spectrum which is given by 2

/M(w)j =

(1-120)

where c p j is the one electron sum of the I

individual momenta, cpo is the ground state, is one of the exciton states whose and general form is given in Eq. (1-110). One and express qo and can expand q(kh ke)as Slater determinants, in which case Mexcbecomes:

veXc

-

vex,

= x(kh/P1ke)a(ke,kh)

where m* is the reduced mass. This sum reduces to the density of states of a 3D electron gas, so that

(1-115)

ke , k h

We have seen that the one particle matrix elements are non-zero for vertical transi-

It is better for comparison with IMexc12to calculate the integrated IM( o)l2 up to a fre-

34


quency o.One thus gets

where E,, is the exciton binding energy in the Is state. Typical values are E , , = 10 meV, h o - E, = 200 meV in which case the ratio, Eq. (1 - 122), is of the order of 20%.

1.3.4 Ab Initio Calculations of the Excitonic Spectrum Ab initio calculations of E ~ ( wfor ) semiconductors have recently been reported (Albrecht et al., 1997). They generalize the G-W approximation of Sec. 1.2.1.4 to the case of electron-hole excitations. This requires the calculation of the two-particle Green's function, which obeys the BetheSalpeter equation (Nozikres, 1964; Sham and Rice, 1966; Strinati, 1984). This is equivalent to solving a Schrodinger equation for the electron-hole pairs with complex potential energy meaning that true eigenstates of the system are not obtained. The dominant potential energy term is the dynamically screened electron-hole attraction, while the smaller exchange term remains unscreened. Again, as for GW, the are0 greatly ) improved over results for ~ ~ ( those obtained in LDA, for which the main peak is incorrectly described (Albrecht et al.. 1997). 1.3.5 A Detailed Comparison with Experiments We give an account here of some results comparing empirical pseudopotential bands with X-ray photo-emission and inverse

photo-emission results (Chelikowsky et al., 1989). These experimental techniques combined with reflectivity data can yield nearly complete information concerning the occupied and empty states. A comparison of experimental data with theory will then provide a stringent test of the validity of the predictions. The empirical pseudopotential used by Chelikowsky et al. (1989) and Cohen and Chelikowsky (1988), was built from a local and a non-local part. The local part was, as described before, restricting the non-zero components to be V,, V,, V8, and V l l . As emphasized by Chelikowsky et al. (1989), this procedure yields and accurate description of the reflectivity and photo-emission spectra. However, for Ge, GaAs, and ZnSe, non-local corrections to the pseudopotential are necessary to produce similar accuracy. This is caused by d states within the ion core which modify the conduction band structure of these three materials and make the pseudopotential non-local. As discussed before, a simple correction for a specific /-dependent term can be written v,,(k,c

4x -~')=-(21+i)fi(cose).

sz,

. jdrr2w-)j,(%)j!(X2

(1-123)

where x = k + G , c o s O = x . x ' l x x ' , Qa is the atomic volume, P, is a Legendre polynomial, j , is a spherical Bessel function, and V,(r) is the non-local correction. In Chelikowsky et al. (1989) a simple Gaussian form has been used for V, whose parameters are fitted to experiment. The local and non-local parameters are tabulated in Chelikowsky et al. (1989). The cut-off in the plane wave expansion was taken at an energy E , = 8 Ry and extra plane waves up to E , = 13 Ry were introduced by the perturbation theory. The corresponding results for Ge and GaAs are compared to experimental

4,r 35


4r

n-Ge(l11)12xl

p-GaAs(ll0) 1 x 1

1

A

r A

X

w K

-10

-5

0

5

10

15

20

z r

-IO

-5

0

5

10

15

20

Energy relative to E, (eV)

Energy relative to E, (eV)

Figure 1-14. Comparison between photo-emission measurements and calculated densities of states for Ge: experimental intensities (top), theoretical density of states (middle), corresponding calculated bands (bottom) for which the vertical axis corresponds to the wave vector. The experimental data corresponds to Xray photo-emission spectroscopy (XPS) or to Bremsstrahlung isochromate spectroscopy (BIS).

Figure 1-15. Comparison between photo-emission measurements and calculated densities of states for GaAs.

data in Figs. 1-14 and 1-15. The agreement is fairly good, especially if one notes that both photo-emission and optical data are reproduced with similar accuracy.

At this point it is also interesting to measure the accuracy of the first principles G-W calculations for these materials. This must be done keeping in mind that these calculations are performed starting from local density calculations which, as we have seen, lead to large discrepancies in excitation energies. One result taken from Hybertsen

36


like structures such as Se or Te for which we use the tight binding description, comparing the results with photo-emission data. We briefly discuss the case of lamellar materials. Finally, we consider a class of semiconductors with unconventional bonding, the Sb chalcogenides.

2

Ge 0

-2

z

-4

1.4.1 General Results for Covalent Materials with Coordination Lower than Four

v

-6 c

W

-a 0

-10

-14

L

Experiment

* Typical error

-

A

r

A

X

Wave vector i; Figure 1-16. Comparison between angular resolved photo-emission and the G-W calculation of Hybertsen and Louie (1986) showing typical experimental errror bars.

and Louie (1 986) compares the valence band structure with angular resolved photoemission (see Fig. 1 - 16).

1.4 Other Crystalline Materials with Lower Symmetry In this section other cases of crystalline semiconductors are examined. In these the bonding is more complex than that in zincblende materials which constitute, in a sense, the prototype of covalent or partly ionic bonding. We begin by generalizing the tight binding arguments discussed for tetrahedral systems to cases with lower coordination. We then specifically consider chain-

Let us consider systems where each atom has N equivalent bonds to its neighbors. As with tetrahedral compounds, we want to build a molecular model which provides a simple basis for the understanding of their band structure. On each atom we build N equivalent orbitals which point exactly or approximately towards the nearest neighbors. We consider in all cases an sp minimal basis: this is always possible since one has four basis states from which one forms only Ndirected orbitals, with N e 4 . The remaining atomic states are then chosen by taken into account the local symmetry (we shall later see specific examples of how this can be achieved). Once this is done the basic electronic structure follows almost immediately. Again the directed states strongly couple in pairs as in diatomic molecules and form o bonds with a 0 bonding state and a a * antibonding state. It is this coupling which dominates the Hamiltonian matrix and the cohesive properties. To the lowest degree of approximation, all other states remain uncoupled at their atomic value. The resulting level scheme then consists of N / 2 abonding and N / 2 o * antibonding states plus 4 - N non-bonding atomic states per atom at energies which depend on the specific case under consideration. Of course, these levels are all strongly degenerate and, if the molecular model is meaningful, further inter-

1.4 Other Crystalline Materials with Lower Symmetry

37

actions will lift this degeneracy to form well-defined and separate energy bands.

1.4.2 Chain-Like Structures Like Se and Te One instructive example of a chain-like structure is pictured in Fig. 1-17, which represents an elemental system where each atom forms two equivalent bonds with an interbond angle of 90”. This situation does not exactly represent the crystalline structure of Se and Te but is very close and will help us in understanding the properties of these materials. We will discuss this case from the tight binding point of view, the same qualitative description of the bands being obtained with the pseudopotential approach. Let us begin the tight binding picture by starting from the molecular model. As discussed before we first have to build two directed states pointing towards the neighbors. With the local axes of Fig. 1-17, these are pure p states px and py with the positive lobe oriented towards the neighbors. Then, from symmetry, the other atomic states which do not participate in 0 bonds will be the s state and the pz state, perpendicular to the plane of the two bonds. The parameters appropriate to Se and Te are such that roughly E p -E, = 10 eV, H,= - 2 eV which from Eq. (1-64) means that H,, = - 1.3 eV and H , = - 0.6 eV. The results of the molecular model are pictured in Fig. 1-18 with the corresponding electron

Figure 1-17. Simplified chain-like structure for Se and Te.

Figure 1-18. Molecular levels for Se and Te, the energy scale being of order 15 eV between the s and u* states.

Energy (eV)

Figure 1-19. Photo-emission results for Se (a) and Te (b) compared to the calculated densities of states: - - - (experiment), - (theory).

population per atom and the nature of the states. The upper valence band is the nonbonding pz band corresponding to the wellknown “lone pair” electrons. Comparison with the photo-emission results of Fig. 1-1 9 (Shevchik et al., 1973) shows that the molecular model already gives an essential account of the results.

38


As for covalent systems, we can now study the broadening of the molecular levels into bands. We treat each band separately, which is valid to the first order in perturbation theory. We make use of a nearest neighbors tight binding Hamiltonian as defined in Sec. 1.2.3.1 with four parameters H,,, H,, H,,, and H,, (note that the H, connecting the px, py functions involved in o bonds are already included in the molecular model). The case of the s band is the simpler one, with an interaction H,, between nearest neighbors. We are dealing with a system containing two atoms per unit cell (Fig. 1- 17) and easily get the s band dispersion relation: E,(k) = E, k 2 H,, ~

ka

~

COS -

(1-124)

L

The pz lone pair band can be treated in the same way since the pz orbitals are coupled This together only via the interaction H,. gives: E,(k) = E, k 2 H,, ~

ka cos -

(1-125)

L

The broadening of the o a n d o* bands is slightly more involved. Let us refer the bonding and antibonding orbitals to one sublattice whose atoms are labelled i, their neighbor in the cell being labelled i'. From Fig. 1-17 the bonding and antibonding states can be written: (1-126)

The energies of these states are flH,,I. With our tight binding Hamiltonian there is no interaction between adjacent bonds but only with second nearest neighbor bonds. Moreover, the x-like bonds do not couple with the y-like bonds giving rise to doubly degenerate bands. It is simple to show that only the H,,interaction is involved and one

gets the following dispersion relations: Eb(k) = - IH,, I + I H , Ea (k) = + I I + I H,,

I cos (k a>

I COS (k a)

(1-127)

Of course the linear chain model of Fig. 1-17 does not exactly correspond to the real structure of Se and Te. These Materials are characterized by helicoidal chains with three atoms per unit cell ( H u h , 1966). The interbond angle is 100" for Se and 90" for Te. Under such conditions the basic features of the previous model still remain valid. When the bond angle is not exactly 90", one builds on each atom two symmetrical px and py orbitals in the plane of its two bonds, pointing only approximately towards its neighbors. One then builds a pz orbital perpendicular to the plane and, finally, the s orbital is left uncoupled. This leads to a molecular model exactly identical to the previous one except that the bonding and antibonding levels will now be at +IH,,I sin2(8/2), where 8 is the bond angle. The dispersion relation will also be slightly modified with respect to those in the simple model, especially for the lone pair band since the pz orbitals are no longer parallel. However, all the basic qualitative features will remain unchanged. All conclusions of the tight binding description are confirmed by experiments (Shevchik et a]., 1973) and also by empirical pseudopotential calculations (Schliiter et al., 1974). The main difference is that the weak interaction between chains induces some degree of three-dimensional characters which wash out to some extent the onedimensional divergences of the density of states (Schluter et al., 1974).

1.4.3 Layer Materials It is not possible here to give a complete account of what has been done on layer materials. We thus restrict ourselves to the


case of the crystalline germanium monochalcogenides Ge-Se and Ge-Te for which we can generalize the simple model discussed above for the Se and Te chains. In these material both Ge and Se (or Te) atoms have threefold coordination with interbond angles close to 90". Furthermore, they consist of two-dimensional layers, see Fig. 1-20. We can generalize the molecular model to this kind of system in a straightforward manner. We idealize the situation by considering interbond angles of 90" and build local axes 0 x, y, z, on each atom along these bonds. The natural basis states for the molecular model will be the s states and the corresponding px, py, pz orbitals, each of these having its positive lobe pointing toward one neighbor. In the molecular model the s states remain uncoupled while all p orbitals couple by pairs, leading to a bonding and a * antibonding states. The resulting level structure is shown in Fig. 1-21. The number of electrons per GeSe unit is 10. As there are 2"s" states, 3 a, and 3 a * states per unit, one concludes that all s and astates are filled and these form the valence band. The fundamental gap then takes place between the a bonding and a * antibonding states. The level position in Fig. 1-21 have been calculated using the parameter values given in Bergignat et al. (1988) for GeSe. The s level of Se is much lower, followed by the Te s level, and the 0 and a * levels. The resulting valence band compares extremely Ge

Figure 1-20. Simplified layer structure for GeSe.

39

0 '

SI@

4

s,se+. Figure 1-21. Molecular levels for GeSe, the energy scale between the s and o* levels being of order 15 eV.

well with the XPS measurements for crystalline GeSe, noting that the densities of states of the alevels are 3 times larger than for s states. 1.4.4 New Classes of Materials: the Antimony Chalcogenides

One major interest of these materials is that they clearly show how band theory can clarify the understanding of the chemical bond in a class of semiconductors with fairly complex structure. This subsection is a summary of what can be found in Lefebvre et al. (1987 and 1988). Tin and antimony atoms have an electronic configuration Sn:[Kr] 4d1°5 s25p2, Sb:[Kr] 4d1'5 s25p3. Some chalcogenide compounds are insulators in which the bonds have strong covalent character while others are characterized by a lone pair, 5s2, which does not take part in the bonding but whose properties are directly correlated to the Sn or Sb coordination and to the structural packing (Gillespie and Nyholm, 1957) (for instance, such a correlation is the basis of chemical valence shell electron pair repulsion VSEPR theories (Gillespie, 1972)). These materials are characterized by a large range of electrical behaviors (insulator, semi-conductor, semi-metal). Their general formula can be expressed as BbX,, A,BbX,, Or A,BbX,Ii where the atoms are alkaline or alkaline-earth or T1,

40


Pb for A, Sn or Sb for B, a chalcogen for X(S, Se, Te), and iodine for I. Until recently, systematic analyses of this family of materials have consisted of a determination of their atomic structure and their electrical conductivity and also of Mossbauer experiments performed on ‘19Sn and on ‘*’Sb.This has allowed Ibanez et al. (1986) to build a simple chemical bond picture using the concepts of “asymmetry” and “delocalization” of the 5s2 lone pair. It is thus interesting to analyze the exact meaning of such notions through the combined use of photo-emission (UPS and XPS) measurements and band structure calculations. We show in the following discussion that this allows us to obtain a coherent picture of the electronic properties of this class of materials and, furthermore, that one can derive a molecular approximation of the full band structure which allows a clear understanding of the physical nature of the distortion experienced by the 5s’ lone pair electron distribution. We consider five representative elements of the Sb family chosen to exhibit the whole range of electronic properties, Le., SbI,, Sb,Te,, SbTeI, TlSbS,, and Tl,SbS,. Let us first summarize the previous understanding of the electronic properties of these materials. First, X-ray diffraction studies have provided the bond lengths and interbond angles which have been connected to the bonding character (covalent, ionic, or Van der Waals), the distortion of the packing around antimony atoms being attributed to the stereochemical activity of the 5s’ lone pair E(Sb). Mossbauer spectroscopy gives information on the electron distribution around the Sb atom through the isomer shift 6 (directly connected to the 5s electron density at the nucleus) and the quadrupole splitting A which reflects the electric field gradient. Using these data plus the electrical properties (conductivity a,

band gap Eg),these materials have been classified into three groups on the basis of their E(Sb) behavior: i) E(Sb) is stereochemically inactive, localized around the Sb nuclei with strong 5s’ character. This corresponds to octahedral surrounding of the Sb atoms and to insulating behavior (ex: SbI,). ii) E(Sb) is stereochemically active as is seen by the distorted surrounding of the Sb atom and the corresponding reduction of the 5s character at the Sn nucleus, which can be attributed to s-p hybridization. These materials are semiconductors (1.2 eV 5 Eg 5 2 eV) with weak conductivity (a=lov6 - IO-* Q-’ cm-’). iii) E(Sb) is again stereochemically inactive, i.e., the Sb environment is again octahedral as in the first group but there is a 5s density loss at the nuclei and these compounds are semi-metals (Eg 0.1 eV, a= 10, R-’ cm-I). The lone pair E(Sb) is then considered as partly delocalized (Ibanez et al., 1986). On the basis of these known properties a tentative description of the band structure has been proposed (Ibanez et al., 1986) whose main features are schematized in Fig. 1-22. Fig. 1-22 (1) corresponds to case (i), (2) and (3) to case (ii) with sp hybridization, and (4)to the delocalized 5s2 lone pair. The important point is that such a description assumes that the “5s” band is moving from the bottom to the top of the valence band which is relatively hard to understand on simple grounds. A better understanding of these electronic properties requires a band structure calculation which is difficult to perform in view of the large numbers of atoms per unit cell (between 6 and 24 for the materials considered here). For this reason it is necessary to use empirical tight binding theory whose simplicity allows a full calculation. The


rp B

d(Sb)

E

Sb atomic levels

1

2

3

tight binding description for these materials is based on the use of an “s-p” minimal basis set and, as usual, on the neglect of interatomic overlap terms. The Hamiltonian matrix elements are taken from Harrison’s most recent set of empirical parameters (Harrison, 1981). However, the situation here is more complex since there are several close neighbor distances. To account for this, we apply Harrison’s prescription to the interatomic matrix elements Hap ( R , ) for atoms which are at the nearest neighbor’s distance R , and determine the other H a p (R) by the scaling law:

I j,“ 11

Hap(R)=Hap(R,)exp -2.5 --1

(1-128)

valid for R lying between R, and a cut-off distance R, chosen to correctly represent the crystal. The use of an exponential dependence and of the parameter 2.5 is discussed in Allan and Lannoo (1983). Of course, any type of empirical theory has to be tested in several ways and, for this reason, we use XPS-UPS measurements to confirm the nature of the predictions for the valence band densities of states for which tight binding should give a reasonable description. Some characteristic results are reproduced

41

Figure 1-22. Initially proposed densities of states for the antimony chalcogenides, the energy scale being 10 to 15 eV between the lower und upper levels.

P 4

in Fig. 1-23 where it can be seen that the empirical tight binding calculation correctly gives the number and position of the main peaks of the valence band density of states and also gives the correct valence band width. In principle, tight binding is better suited to the description of valence bands but Table 1-2 shows that the predicted gaps also compare well with the experiments. One can then calculate the number N s of 5s electrons on the Sb atom which is the basic quantity determining the Mossbauer isomer shift 6. Fig. 1-24 shows the linear correlation between the measured S and the computed values of N,. This proportionality demonstrates that the band structure determination gives a completely coherent picture of the electronic structure of these compounds. A striking feature of these results is that the loss of 5s electrons is extremely small with a maximum value of about 0.1. The notion of strong or weak 5 s character on Sb is thus quite relative as is the notion of delocalization since the band states are always delocalized over the whole crystal. However, if we sum up their contribution to the 5s population on the Sb atoms we get something that is always close to N , = 2. A final comment that one can make concerning the band structure is that in all cases one finds the 5s Sb density of states at the bot-

42


2-N, (calc) 0.05 0

Energy (eV)

Figure 1-23. Predicted valence band densities of states for antimony chalcogenides compared to UPS and XPS spectra. The full lines correspond to theory. The vertical axis represents intensities or densities of states in arbitrary units.

Table 1-2.Predicted gaps (E,, p ) compared with experimental gaps (Eg,e ) in eV. Compound

E,, P

E,, e

SbI, Sb2Te, SbTeI TISbSz TI,SbS,

2.40 0.14 1.32 1.73

2.30 0.2 1 1.45 1.77 1.80

2.12

tom of the valence band in contradiction with the qualitative chemical picture of Fig. 1-22 that relates the 5s2 delocalization to a shift of this density of states towards higher energies.

0.10

*

i““. Figure 1-24. The Sb Mossbauer isomer shift 6 versus 2 - N s ( N , being the number of Sb s electrons) and the number of N of missing neighbors.

At this level, a better understanding can only be obtained using a simple physical model. To do this we idealize the atomic structure by considering that a reasonable first order description of the immediate environment of an Sb atom consists of the perfect octahedron of Fig. 1-25. However, among the 6 possible sites i = 1 to 6, N are taken to be vacant. We then treat such a Sb-M6-N unit as a molecule for which we take as basis functions: cps, cpx, ‘py, and cpz which are the s and p states of the Sb atom and xiwhich are the “p” states of the existing M atoms that point towards the Sb atom (i.e,, their positive lobe is in its direction). The “p” levels of the Sb and M atoms fall in


I,'

p -$.---

--

I

0'1

pa

06

3-N

Sb

X

P

10

'\\-

Xi

tZ

3-N

*. \ I\\ ' -

s

----__-

43

Figure 1-25. Simplified molecular model of the antimony chalcogenides: a) idealized octahedron around one Sb atom, b) the two possible situations along one axis with no or one missing neighbor, c) the resulting schematic level structure.

Y

the same energy range and, to simplify, we take them as degenerate while the s level of the Sb atom is about 10 eV lower. For this reason we first treat the coupling between the p states alone. For a given direction a = X, Y, or Z two situations can occur (see Fig. 1-25b): i) There are two M atoms i(=l, 2, or 3) and j ( = 4, 5 , or 6). In this case the Sb p state qa only couples to the corresponding -x j ) / f l antisymmetrical combination giving rise to a o bonding state and a o* antibonding state while the symmetrical (xi+ remains at the p energy. ii) There is only one M atom i. In this case (pa couples to to give again a o* and o state but with a smaller splitting.

(xi

xj)/*

xi

The resulting level structure is shown in Fig. 1-25c. All these molecular levels have to be filled except for the o* antibonding levels. At this stage the 5s electron population on the Sb atoms is exactly N s = 2. The only factor that allows for a reduction in N s is the coupling of qs with the empty antibonding states 0". However, in case i) these are antisymmetrical so that the coupling vanishes by symmetry. This is not true in case ii) where such coupling will exist and will induce a reduction in N,. We then arrive at the conclusion that the total reduction in

N s will be proportional to the number N of missing M atoms, if O S N S 3 . Fig. 1-24b shows that the plot of the chemical shift S versus N also gives a straight line but with a reduced accuracy as compared to Fig. 1-24a. The constant factor relating 2-N, to N can be evaluated by the second order perturbation theory and gives a result comparable to Fig. 1-24 a. This means that the simple molecular model contains the essence of the behavior of the 5s lone pair. The combination of XPS measurements and band structure calculations thus leads to a fully coherent description of these materials allowing us, for instance, to understand the trends in the Mossbauer chemical shift related to the 5s electron density at the Sb nucleus. This gives a precise meaning to the empirical notions of asymmetry and delocalisation of the 5s2 lone pair which were previously used in solid state chemistry. Similar considerations were then applied to analyze the meaning of the chemical notion of tin oxidation number in the compounds belonging to the SnS-In2S,-SnS2 family (Lefebvre et al., 1991). The conclusion is that the difference between the two oxidation states Sn(I1) and Sn(1V) corresponds to a variation of about 0.7 Sn 5s electrons, the SnII atoms always being in strongly distorted sites. The ETB technique

Next Page

44


is equally helpful to predict trends in the valence states of binary and ternary gallium and arsenic oxides and compare them with photoemission data (Albanesi et al., 1992). A particularly interesting application is the case of lithium insertion in three dimensional tin sulfides (Lefebvre and Lannoo, 1997), where the change in tin oxidation state was related to the fact that the lithium donor turns out to be a defect with negative correlation energy. Finally, the family of tin monochalcogenides from SnO to SnTe was also studied by ETB combined with abinitio LDA calculations, both leading to similar conclusions (Lefebvre et al., 1998).

1.5 Non-Crystalline Semiconductors In the preceding section we have discussed the properties of several crystalline covalently bonded systems with varying coordination numbers. These are usually determined in such a way that each atom satisfies its local valence requirements. In most cases this leads to the 8 - N or octet rule which links the coordination Z to atomic valence through the relation Z = 8 - Nfor N 2 4.However, there are a lot of exceptions to this rule, for example, the case of crystalline Ge-Se i n which 2 is 3 for both Ge and Se instead of being respectively 4 and 2. In all these covalently bonded materials the cohesive energy mainly results from the formation of the local bonds between nearest neighbors and not from incomplete filling of a broad band as in metals. This cohesive energy is much less sensitive to variations in interbond angles, and to long range order than to stretching of the covalent bonds. This explains why, under appropriate preparation conditions, most of these materials can be found either in the amorphous or in the glassy state character-

ized by a loss of the long range order. However, as shown by the determination of their radial distribution function, these systems still possess a well-defined short range order similar to what is observed in the crystalline phases. In this section we examine some features of non-crystalline semiconductors. We first consider some elemental amorphous systems like a-Si, a-As, and a-Se and examine possible modifications in the density of states. We then detail the properties of a typical intrinsic defect likely to be present in aSi, i.e., the isolated dangling bond. Finally we make some comments on the electronic structure of more complex systems like Ge,rSel-, ... 1.5.1 The Densities of States of Amorphous Semiconductors

From our qualitative discussion in Sec. 1.1 remember that it is likely that amorphous semiconductors will give rise to energy bands. In these cases, the pseudocontinuum of states arises no only because there are a large number of atoms but also because there is some disorder inherent to such structures which tends to spread the energy spectrum, leading to band tails. To get a more precise feeling for what happens in the amorphous state, it is necessary to build idealized models which could be mathematically tractable and be considered as reference situations. It is for such reasons that continuous random networks have been developed to model systems like a-Si, a-Ge, at a-SiO,. These are constructed by representing atoms and bonds as balls and sticks and connecting them together randomly without loose ends or dangling bonds. Usually such networks lead to a predicted radial distribution function relatively close to the experimental one. However, they remain idealized descriptions since the

Previous Page

1.5 Non-Crystalline Semiconductors

real material can contain clusters, dangling bonds, and other eventual deviations. Assume for a moment that we are dealing with such an idealized lattice for a-Si. The first problem that arises concerns the general shape of the band structure: does it lead to a fundamental energy gap and to the same structures in the valence band as in the crystalline system? An interesting answer to such complex questions can be obtained via simplified Hamiltonians such as those offered by the empirical tight binding theory. This was achieved by Weaire and Thorpe (1971) on the basis of an Hamiltonian first proposed by Leman and Friedel (1962). This was based on the use of sp3hybrids and included only two parameters: the intrabond coupling between such hybrids and the coupling between any two hybrids centered on the same atom. Here we shall reproduce the same conclusions by using the more involved Hamiltonian described in Sec. 1.3.1.1 for crystalline Si. We again build bonding and antibonding orbitals which are exactly the same as for the crystal. We treat the broadening of the bonding and the antibonding band separately. Concentrating on the bonding band, we get the analog of Eq. ( 1 -76), (E-E,+2d)U,,j=d(Si+Sj)

(1-129)

in view of the fact that we are dealing with an elemental system (AA= A, = A). We directly sum this expression overj and get ( E - & - 2 A ) S i = A c Sj

(1-130)

j

where the sum overj extends over the nearest neighbors of atom i. Equations (1-130) are the same as those one would obtain for an s band on the same lattice with unit nearest neighbors interactions

&Si = E sj j

(1-131)

45

The eigenvalues of the two matrices are thus related through E = E,

-t 2

4 + A&

(1-132)

The Hamiltonian matrix defined by Eq. ( 1 -13 1 ) is known as the connectivity matrix and general topological theorems (Ziman, 1979) allow us to show that its eigenvalues E must lie in the interval [- 4,+ 41 imposed by the coordination number. This means that the energies in the bonding band must lie in the range (1-133) E,+ 2 4 - 4 [AI S E E E b + 2 A + 4 ]AI This is the same result as for the crystalline case and thus shows that the bonding band for this idealized model of a-Si must be contained in the same energy interval as for cSi. The same reasoning applies to the antibonding band. Thus the gap between these two bands is at least as large as for the crystal in the same model. Inclusion of the interactions between the bonding and antibonding states can only increase this gap by mutual repulsion of the two bands. The first conclusion of the model is thus that the gap still exists in a-Si. The second question concerns the structures in the valence band. In the crystal some characteristic structures of the density of states are due to the Van Hove singularities which are a signature of long range order, as we discussed above. In the amorphous system these should be washed out as shown by the comparison of the density of states I Z ( E ) of the connectivity matrix Eq. (1-131) between the crystal and a Bethe-lattice with coordination 4 (Lannoo, 1973). This is efcurves 0 ) fectively what the XPS and ~ ~ ( represented in Fig. 1-26 show for c-Si and a-Si. The real atomic structure in a-Si is likely to deviate from the idealized one. To satisfy the local bonding constraints, distortions in bond angles and even in bond

46


(1 979) and detailed consideration about localized and extended states can be found in Ziman (1979) and Weaire (1981).

40 c-SI

30 -

A

w"

20

-

10

-

1.5.2 Numerical Computations

1 I

-

3

I

\!

Improvements in computational power have now allowed numerical calculations of the electronic states in amorphous structures. Major progress in this regard has been the derivation of computer-generated random continous networks with tetrahedral

0

lengths are likely to occur. These will give rise to band tails. It is thus expected that the gap region will not be free of states since the conduction and valence band tails will overlap as shown schematically in Fig. 1-27. As regards the transport properties, the localized or the delocalized nature of these states is of importance. A commonly adopted view (Ziman, 1979: Weaire, 198 1 ) is that of a mobility edge separating the two types of states as shown in Fig. 1-27. Finally, the situation is complicated further by the existence of defects i n the bonding, in particular the isolated dangling bonds which are treated in the following section. A more complete review of amorphous or glassy semiconductors is given in Ziman

E,

E,

Figure 1-27. Schematic density of states in the gap region of amorphous semiconductors.

et al., 1988), the first order Raman spectrum is found to be in very good agreement with the experimental one (Marinov and Zotov, 1996). It also reproduces with good accurracy the experimental radial distribution function of amorphous silicon. Recent theoretical studies of phonons in a-Si also confirm the superiority of this model compared to others (Knief et al., 1998). The electronic structure for the WootenWiner-Weaire model has been calculated (Allan et al., 1998) in the sp3s* ETB approximation (Vogl et al., 1983) and a d-2 variation of the tight binding parameters with distance d (Harrison, 1980). Application of the method to a-Si gives rise to large band tailing, which is attributed to a small number of bond angles that deviate greatly from the tetrahedral 109" value. An efficient simulation of hydrogenated amorphous silicon (a-Si : H) was obtained by removing the atoms with the most distorted bond angles and saturating the resulting dangling bonds with hydrogen atoms. The resulting gap and Urbach tails then become comparable with the experimental one (Allan et al., 1998). The same description


has allowed the study of confinement effects for quantum dots of a-Si-H (Delerue et al., 1996) and to show that doping of a-Si-H results in much larger binding energies of donors and acceptors than in the crystalline material (Allan et al., 1999).

1.5.3 Dangling Bonds A lot of physical situations in tetrahedrally coordinated materials involve the rupture of bonds. The simplest well-documented case is the P, center at the Si-SO2 interface which corresponds to a tricoordinated silicon atom, i.e., to the isolated dangling bond. Such defects are also likely to occur in a-Si. Another well-known situation is the vacancy in silicon (and to a less extent in compounds) where there are four interacting dangling bonds. Let us first shortly recall the basic physical properties of dangling bonds. The simplest description comes from a tight binding picture based on an atomic basis consisting of sp3 hybrid orbitals. The properties of the bulk material are dominated by the coupling between pairs of sp3 hybrids involved in the same nearest neighbor’s bond. This leads to bonding and antibonding states which are then broadened by weaker interbond interactions to give, respectively, the valence and conduction bands. In the bonding-antibonding picture, the rupture of a bond leaves an uncoupled or “dangling” sp3 orbital whose energy is midway between the bonding and antibonding states. When one allows for interbond coupling, this results in a dangling bond state whose energy falls in the gap region and whose wave function is no longer of pure sp3 character, but is somewhat delocalized along the backbonds. Experimentally this isolated dangling bond situation is best realized for the P, center, i.e., the tricoordinated silicon atom

47

at the Si-SiO, interface (Poindexter and Caplan, 1983; Caplan et al., 1979; Johnson et al., 1983; Brower, 1983; Henderson, 1984) but it can also occur in amorphous silicon (Jackson, 1982; Street et a]., 1983) as well as in grain boundaries or dislocations. It has been identified mainly through electron spin resonance (ESR) (Poindexter and Caplan, 1983), deep level transient spectroscopy (DLTS) (Johnson et al., 1983; Cohen and Lang, 1982) and capacitance measurements versus frequency and optical experiments (Jackson, 1982; Johnson et a]., 1985). The following picture emerges: i) The isolated dangling bond can exist in three charge states: positive D’, neutral Do, and negative D-.These respectively correspond to zero, one, and two electrons in the dangling bond state. ii) The effective Coulomb term UeE,Le., the difference in energy between the acceptor and donor levels, ranges from =0.2 to 0.3 eV in a-Si (Jackson, 1982) to about 0.6 eV at the Si-Si02 interface (Johnson et al., 1983). The ESR measurements give information on the paramagnetic state Do through the g tensor and the hyperfine interaction. Their interpretation indicates that the effective “s” electron population on the trivalent atom is 7.6% and the “p” one 59.4%, which corresponds to a localization of the dangling bond state on this atom amounting to 67% and an “s” to “p” ratio of 13% instead of 25% in a pure sp3 hybrid. This last feature shows a tendency towards a planar sp2 hybridization. Several calculations have been devoted to the isolated dangling bond. However, only two of them have dealt with the tricoordinated silicon atom embedded in an infinite system other than a Bethe lattice. The first one is a self-consistent local density calculation (Bar-Yam and Joannopoulos, 1986) which concludes that the purely electronic value of the Coulomb term (Le., in the

48


absence of atomic relaxation) is U = 0.5 eV. The second one is a tight-binding Green's function treatment in which the dangling bond levels are calculated by imposing local neutrality on the tricoordinated silicon (Petit et al., 1986). In this way the donor and acceptor levels are respectively ~ ( 0+), = 0.05 eV and ~ ( 0-), = 0.7 eV. Their difference corresponds to U = 0.65 eV, which is in good agreement with the local density result. Both values correspond to a dangling bond in a bulk system and can be understood simply in the following way: the purely intra-atomic Coulomb term is about 12 eV for a Si atom: it is first reduced by a factor of 2 since the dangling bond state is only localized at 70% on the trivalent atom; finally, dielectric screening reduces it by a further factor of E = 10. The final result 6/e gives the desired order of magnitude 0.6 eV. At the Si-SiO, interface, however, the situation becomes different because screening is less efficient. A very simple argument leads to the replacement of E by ( E + 1)/2 so that the electronic Coulomb term for the Pb center should be twice the previous value, i.e., u(Pb) = 1.2 eV. An extremely important issue is the electron-lattice interaction. There is no reason for the tricoordinated atom to keep its tetrahedral position. A very simple tight binding model (Harrison, 1976) shows that this atom does indeed experience an axial force that depends on the population of the dangling bond state. This is confirmed by more sophistical calculations (Bar-Yam and Joannopoulos, 1986). The net result is that, when the dangling bond state is empty (Df) then the trivalent atom tends to be in the plane of its three neighbors (interbond angle 120"). On the other hand, when it is completely filled (D-) it moves away to achieve a configuration with bond angles smaller than 109" as for pentavalent atoms. Finally, the situation for Do is obviously intermediate

with a slight motion towards the plane of its neighbors. For the three charge states D+, Do, and D-, corresponding to occupation numbers n = 0, 1, and 2, respectively, one can then write the total energy in the form E(n,u) = = n Eo + (1/2) U n2- F ( n ) * u

( 1- 134)

+ (112) k u2

where u is the outward axial displacement of the tricoordinated silicon atom, F ( n ) the occupation dependent force, U the electronelectron interaction, and k the corresponding spring constant which should show little sensitivity to n. We linearize F ( n ) F ( n ) = FO

+ Fl

(TI

-

1)

(1-135)

an minimize E(n, u ) with respect to u to get Emi,(n). The first order derivative of Emi,(n)at n = 1/2 and n = 3/2 gives the levels ~ ( 0+), and E ( - , 0). The second order derivative gives the effective correlation energy: Fi2 U,ff = u - k

(1-136)

Theoretical estimates (Bar-Yam and Joannopoulos, 1986; Harrison, 1976) give F1= 1.6 eV k'and k = 4 eV (A2)-' (Lannoo and Allan, 1982) so that F:/k is of the order 0.65 eV. This has strong implications for the dangling bond in a-Si where Ueff becomes slightly negative as concluded in Bar-Yam and Joannopoulos (1986) but this result should be sensitive to the local environment. On the other hand, with U = 1.2 eV, the P, center at the Si-Si02 interface would correspond to U,,-0.6 eV, in good agreement with the experiment. The theoretical finding that U,, is slightly negative for the dangling bond in a-Si leads to an inverted order for its levels, in which case the Do state could never be stable (Bourgoin and Lannoo, 1983). This


does not agree with the experiment, in which an EPR spectrum which seems characteristic of Do has been observed. One possible reason for this discrepancy is the suggestion that a-Si may contain overcoordinated atoms (Pantelides, 1986) which might be responsible for the observed features. However, recent careful EPR measurements (Stutzmann and Biegelsen, 1989) seem to rule out this possibility, practically demonstrating that dangling bonds indeed exist and with a positive Uefi This would mean that theoretical calculations have underestimated the electronic U for reasons which are still unclear.

1.5.4 The Case of SiO, Glasses We now give a simplified analysis of the valence band structure of these glasses based on an extension of the tight binding arguments developed before. Let us first consider the case of S O z . The molecular model is essentially the one developed by Harrison and Pantelides (1976). The building Si-0-Si unit is shown in Fig. 1-28. Again one builds sp3 hybrids on the Si atoms while on the oxygen atom one keeps the natural sp basis. The oxygen “s” state is by far the lowest in energy and, to

49

the first order, its coupling with other states can be neglected. It will remain atomic-like at its atomic value E,(O). On the other hand, the oxygen p energy E p ( 0 ) is closer to the silicon sp3 energy l? and the interaction of the corresponding states must be taken into account. The molecular states of the Si-0-Si unit of Fig. 1.28 are then built from the two sp3 hybrids a and b pointing towards the oxygen atom and the px, py, and pz oxygen “p” states. It is clear that pz, being perpendicular to the Si-0-Si plane, will remain uncoupled at this level of approximation giving one state at the atomic value Ep(0).Thus the sp3 states a and b will only couple to px and py via the projection of these states along the axis of the corresponding nearest neighbor direction. All interactions reduce to one parameter ppdefined as the interaction between an sp3 orbital and the p orbital along the corresponding bond. By symmetry px only interacts with (a-b)/* giving rise to strong bonding and antibonding states at energies (1-137)

while py and (a + b)/@ lead to weak bonding and antibonding states (1-138) L

U Figure 1-28. Si-0-Si unit for building the molecular model with the two sp3 hybrids of the Si atoms and the three p states of the 0 atom, p z being perpendicular to the plane of the figure.

where 2 8 is the Si-0-Si bond angle. The resulting valence band density of states per Si-0-Si unit is pictured in Fig. 1-29. It consists of delta functions at energies E,(O), ESB, EWB, and Ep(0), the

50


Figure 1-29. a) SiOz density of states calculated by Robertson (1 983). b) Density of states in the molecular model with a Gaussian broadening of 0.5 eV. - T h e vertical axis corresponds to densities of states in arbitrary units, the horizontal axis to energies, in units of 5 eV, SB, WB and LP denote strong bonding, weak bonding and lone pair states.

-25

-20

-15

-10

-5

0

weight of each state per Si-0-Si unit being equal to unity. The influence of further interactions can now be analyzed as for pure Si. If we call A, and A, the interaction between strong and weak bonding states belonging to adjacent Si-0-Si units then we can repeat the treatment previously applied to Si simply by replacing A with A, or A,. This means that we get densities of states in the strong and weak bonding bands that have exactly the same shape as for Si, consisting of the superposition of a broad and a narrow. almost flat band. This behavior is apparent in the calculated density of states in Fig. 1-29a. These results are in good qualitative correspondence with photo-emission data (Hollinger et al., 1977; Di Stefan0 and Eastman, 1972; Ibach and Rowe, 1974) and more sophisticated numerical calculations (Chelikowski and Schluter, 1977; O’Reilly and Robertson, 1983). Essential information provided by Eqs. ( 1 - 137) and ( 1- 138) is that the splitting between the strong and weak bonding bands is a very sensitive function of the Si-0-Si angle 2 8.Any cause of randomness in Bsuch as the existence of a strained SiO, layer is then likely to induce some broadening of

5

10

these two bands and partially fill the gap between them. Note that in an extreme situation where 8 = 90°, like in GeSe,, the strong and weak bonding energies become identical, leading to a qualitative change in the shape of the density of states. We are now in the position to discuss qualitatively the behavior of the SiO, layer as a function of composition. Numerical calculations have been performed that are all based on a tight binding treatment combined with a more or less refined cluster Bethe lattice approximation (Lannoo and Allan, 1978; Martinez and Yndurain, 1981). Again the molecular model gives precious information that is confirmed by the full calculation. For this, let us consider the results of Lannoo and Allan (1978) which are pictured in Fig. 1-30. In the molecular model each Si-0-Si bond corresponds to a density of states as given in Fig. 1-29b, while each Si-Si bond corresponds to one bonding state which, for the parameters corresponding to Fig. 1-30, falls at an energy slightly higher than E,(O), the energy of the SiO, non-bonding state. If we start from the SiO, limit each Si-Si bond acts like an isolated defect giving rise to a defect level at

1.6 Disordered Alloys

51

energy. At still higher Si concentrations there will be three and then four Si-Si bonds connected to each Si atom. For reasons discussed above such situations are characterized by a flat band at + 2 A and a broad band at lower energies. This regime will thus exhibit a peak of increasing height at the energy of the top of the pure Si valence band. This is exactly what happens in Fig. 1-30. Note also that the strong bonding band has exactly the reverse behavior: the height of its peak and its width both decrease. The last point that is clearly seen in the figure is that, for smaller 8, the separation between the strong bonding states and the weak and non-bonding states is smaller. All these conclusions are in qualitative agreement with experimental observations concerning the valence band of the SiO, systems and of the transition layer at the Si-SiO, interface.


I

-15

-12

-9

-6

-3

E (W

Figure 1-30. Theoretical density of valence band states for S O , systems for different compositions and different bond angles. The flat parts correspond to the zero of n ( E ) in each case.

the energy of the bonding state of Si, i.e., just above the SiO, valence band. When the concentration of Si-Si bonds increases this defective state will begin to broaden into a band with no defined structure at small concentrations. This corresponds to Si04,3 in Fig. 1-30. It is only when the concentration is high enough for Si chains to appear that the situation changes qualitatively. If, as before, A characterizes the interaction between two adjacent Si-Si bonding states, then the DOS of a Si-Si chain exhibits two divergences at + 2 A from the bonding

Compounds with a well-defined lattice but where there is substitutional disorder on the lattice sites compose a series of important electronic structure problems. This is, for instance, the case of pseudobinary semiconductor alloys like Ga,Al,-,As, In,Gal-,As ... . Since such alloys are of much importance, here we give an account of the work performed on these systems. However, before doing this we must introduce general methods like the virtual crystal approximation (VCA), the average t-matrix approximation (ATA), and the coherent potential approximation (CPA). Again these methods, which correspond to complex systems, are applied within the framework of the tight binding approximation. To perform such calculations on disordered systems one must use the Green’s

52


function formalism. For this we introduce the resolvent operator G ( E )of the system, defined as

where H is the Hamiltonian of the system and E is the energy. One major property of the resolvent operator is that the density of states n ( E ) can be expressed as 1 n(E)=--ImTrG(e)

n

Im(G(E))

We want to present different possible levels of approximation here and illustrate them on a tight binding model of a random alloy. This model will consist of a tight binding s band designed to treat a random alloy A,B I --x, the atomic sites forming a lattice with one site per unit cell. This tight binding Hamiltonian can thus be written:

(1-140)

where Tr stands for the trace and Im means imaginary part. When considering a large disordered system, the fact that one takes the trace means that one performs an average over different local situations. One would get the same result by performing an ensemble average, i.e., by considering all configurations that the system could take and weight them by their probability. This means that n ( E ) can also be obtained by per) the resolvent forming the average ( G ( E ) of operator and writing 1 n(E)=--Tr n

1.6.1 Definitions of the Different Approximations

(1-141)

where V is a nearest neighbor interaction taken to be independent of disorder while the on-site terms E/ can take two values; E* for an A atom, for a B one. Let us begin with the simplest type of approximation: the VCA (virtual crystal approximation). As indicated by its name, this consists of assuming that the average Hamiltonian (H) = f gives a correct account of the electronic structure. This means that the exact ( G ( E ) )is replaced by an approximate expression @ E ) :

G(E)= ( E + i q - R1-I

(1-144)

From its definition, the expression of R i n the simple tight binding model is given by:

The advantage is that the quantity ( G ( E ) ) H = [ x + (1 - x > ~~1 C 1 I ) ( I + becomes statistically homogeneous, i.e., if I one has a random alloy on a given lattice, (1-145) +v 10 (1'1 (G( E ) ) acquires the lattice periodicity, while 11' the original G ( E )has not. It is to be noted that ( G ( E ) )is not equal to ( E + i q - (H))-'; which can be solved by using the Bloch theorem. instead we define a self-energy operator We now want to go a step further and do Z ( E )such that one can write the proper averaging of (G) over one site (G( E ) ) = ( E + i q - (H) - ,E( E ) ) - ' ( 1- 142) while the rest of the crystal is treated in an average field approximation. The first obviThe determination of this Zis the aim of the ous thing to do is to start from the virtual following different approximations. crystal, determine G,then consider the local fluctuations on a given site as perturbations and perform the average (GII)for that site,

1

c


where G,, is the diagonal element of G. Let us illustrate this with our specific model. We thus start from fi defined in Eq. (1-145) and look at the possible fluctuations at site 1. If site 1 is occupied by an A atom, the on site perturbation will be WA

= &A - [x &A + (1 - x )

&B]

while for a B atom it will be w B = &B -

A comparison between the two expressions allows us to express ZATA in the form

(1-151)

= (1-146)

= (1 - x ) (&A - &B)

53

[x &A + (1 - x ) &B] = - x(&A-&B)

It is customary to introduce the average t matrix f, equal to the numerator in Eq. (1- 15 1). In this way one gets the usual form of ,Z"TA (see Ziman, 1979, for details):

(1-147) We treat both cases at the same time by assuming that there is a perturbation W , on site 1 with possible values WA or WB. The diagonal matrix element Gll can be obtained by applying Dyson's equation (G = G + G W G ) ,

(1-148)

since G has translational periodicity. We now say that the average (GII) with GI, given by Eq. (1-148) represents a good approximation. This defines the average t-matrix approximation which can be written /

=

On the other hand, (G) is quite generally through Eq. (1-142). To the related to E(&) same degree of approximation as before, one should thus write that (GlJ, is due to the on site perturbation Z'TA applied to site 1. This leads to -

1- ~

This provides a considerable improvement on the VCA, but a definitely better approximation is given by the CPA (coherent potential approximation) which we will now discuss. The CPA uses the same basic idea as the ATA, Le., it is a single site approximation. However, instead of using an average medium corresponding to the average Hamiltonian 6 it is certainly much better to use Z?+2(&), where the 2 is the unknown selfenergy. Starting with this, the perturbation at site 1 is now given by W, - Z ( E )W , , being defined as before. Thus the average (G,,) is given by

(1-153)

\

(1-149)

(GII)ATA=

(1-152)

A T A Goo

( 1- 150)

But one must note that the self-energy is assumed to be the same on each lattice site so that the average crystal resolvent is simply coo, where E is shifted by the coherent potential Z(E).On the other hand, for 2 to be consistently defined, (Gl[)CpA should -Z(E)).Applying this to be equal to GOO(& Eq. (1 - 153) one immediately gets the condition (1-154)

54


which defines the expression of (Ziman, 1979).

1.6.2 The Case of Zinc Blende Pseudobinary Alloys Here we summarize fairly recent calculations performed on these disordered alloys specifically, In, -,Ga,As and ZnSe,Tel (Lempert et al., 1987). This work makes use of an extension of the CPA called the MCPA (molecular coherent potential approximation) which is particularly well adapted to these systems. This case of alloys is of technological interest and it is important to be able to treat disorder effects accurately. This disorder can be conveniently divided into chemical and structural components. The former is related to the different atomic potentials of the two types of atoms, while the latter is associated with local lattice distortions, essentially due to differences in bond length. Such an effect was observed by Mikkelsen and Boyce (1982,1983) in EXAFS (extended X-ray fine structure) measurements on In, -,Ga,As. There the In-As and Ga-As bond lengths were found to vary by less than 2% from their limiting perfect crystal values, despite a 7% variation in the average X-ray lattice constant. This was also found in other zinc blende alloy systems (Mikkelsen, 1984). The electronic structure of these materials is described in the tight binding approximation (extended to second nearest neighbors in the particular calculation of Lempert et al., 1987). The structural problem in an A;-,yA:B alloy causes a difficulty since it leads to local distortions with respect to the average zinc-blende structure. This is overcome by assuming that the atoms lie on the sites of the average crystal but scaling the dominant nearest neighbor interaction to the value appropriate to an A’B or A”B

bond. In such a way the Hamiltonian is defined on a perfectly regular lattice, but even in this case it is not approximation in view of the existence of several orbitals per atom and of diagonal and non-diagonal disorder. The possibility of performing an MCPA calculation is related to the choice of the particular unit cell shown in Fig. 1-31. As in the previous sections, use is made of a sp3 energy E,, the coupling VIA between two different sp3 hybrids, and the intra bond As we have seen, such terms coupling dominate the Hamiltonian for zinc-blende semiconductors and they are likely to give the dominant contribution of disorder effects. As explained at the beginning of this section the disordered alloy Hamiltonian will be replaced by an effective Hamiltonian:

VtB.

&(E)

= (H)+ Z:(E)

( 1 - 15 5 )

MCPA is based on the assumption that the self-energy Z ( E )is cell-diagonal (the cell being defined in Fig. 1-31). It is thus represented by an 8 x 8 matrix in the basis of the

Figure 1-31. The unit cell in the molecular coherent potential approximation, defining the relevant interactions between sp3 hybrids.


sp3 orbitals. With this one can obtain exactly the same expression as in ordinary CPA except that one must replace all quantities W,, Go by 8 x 8 matrices within the same sp3 basis. The consistency condition (Eq. (1 -1 53)) also applies but with matrix inversion and multiplication. We can rewrite it for our two component system (atoms A' and A") as: { [( WAJ-

a GO]-' - 1

+ { [( WA,* -

]-I

+

GO]-'

-

(1-156) 1 }-' = 0

which can be transformed to:

+ ( W A , * - Z ) - ' = 2Go (1-157) or, since WAt + WA,! = 0, to: ( 1- 158) z(E) = - [ WA) - z(E)] C O O ( & -E( E)) [ WAft - z(E)] (WA?-Z',)-'

*

*

*

which is the basic consistency condition allowing the determination of the complex self-energy matrix Z(E). Equation (1- 158) is then solved iteratively to get the unknown Z(E)matrix. In practice, advantage can be taken of the tetrahedral symmetry of the unit cell. The eight sp3 orbitals of this cell can be transformed into basis states for the irreducible representations of the T, symmetry group, i.e., A, and T2 in the present case. On the A site this comes back to the s and p atomic orbitals, while on the neighbors this gives the same combinations of the four sp3 orbitals as the s and p on the central atom. The matrix Eq. (1-158) is then split into four independent 2 x 2 blocks, which considerably reduces the difficulties. It is interesting to give the explicit form of the perturbation matrices WAe and WA,>. We have seen that chemical disorder is modeled by three distinct matrix elements VIA, and V,, in Fig. 1-3 1, which for an atom A lead to an 8 x 8 matrix which we label hA. The average crystal is then given

(for the compound A;_,A:B)

f i =~ (1 - X ) hA' + X ~ A "

55

by: (1-159)

and the perturbation matrices WAf and are: WA! = h A t - hA = x (hA' - hA") (1- 160)

WATt

and

WAU= hA- - h~ =

- (1 - X)

(1-161) ( h ~-r h ~ " )

They are thus defined in terms of the differ- EA,,, V I A t - VIA,,, and v 2 A r B ences "2A"B

*

We will not discuss here the details of the parametrization scheme used in ref. (Lempert et al., 1987), but instead summarize some of their results. Figures 1-32 and 1-33 give the densities of states, comparing the results of the VCA, the site CPA with chemical disorder only, and the MCPA. Clearly the VCA is unsatisfactory for some of the spectral features, particularly for ZnSeo,,Teo.,. A fairly important physical quantity is the band gap bowing, i.e., the deviation from linearity with concentration of the alloy band gaps. The results are shown in Table 1-3 demonstrating that theory does not give a full account of the experimental value of the bowing effect. The difference could be attributed to several causes: uncertainties in the perturbative tight binding parameters, intercell disorder effects, clustering, etc. In any case, it must be realized that the bowing effect is very 3 small, ~ 0 . eV. It is also of interest to briefly discuss other calculations performed on these alloys and compare them with the MCPA. The first of them is the bond-centered CPA of Chen and Sher (1981) which includes disorder effects associated with each bond, Le., in the form of a 2 x2 matrix in the corresponding bonding-antibonding state basis. This includes structural disorder due to VIA and should be, in principle inferior to the MCPA

56


7.0 (a) VCA 3.5 -

- (b) Site-CPA

\

v)

Q,

+iG

v) v

g

3.5 2 -

+

m

c

-

v)

b .2 rA C

-

I

-

+-.

0

0

a

n X

U

K

X

3.5

-

Y

0

I

-12

-0

-4

0

-12

-6

0

6

Energy (eV)

Energy (eV)

Figure 1-32. Density of states in VCA, CPA and MCPA for In,,,Gk,,As. For a discussion see text.

Figure 1-33. Density of states in VCA, CPA and MCPA for ZnSe,,,Te,,,. For a discussion see text.

from that point of view. A second approach is the supercell method which models an Ab,,Ag,,B zinc-blende pseudobinary alloy as a chalcopyrite crystal (Zunger and Jaffe, 1983). This allows them to perform selfconsistent local density calculations. However, the relation to a disordered alloy is not trivial and, for this reason, a calculation has been performed in Lempert et al. (1987) with the same tight binding Hamiltonian in the supercell approach. We reproduce the results in Figs. 1-34 and 1-35 which demonstrate that the supercell results are much closer to the MCPA than are those of the bond-centered CPA. This shows that the main features of the density of states are primarily related to local bonding properties. We should finally mention the recursion method (Haydock, 1980) which should, in principle, be well adapted to this kind of

problem. It allows a direct determination of the local Green functions. This would be fairly efficient for getting the shape of the density of states, but much less efficient for the determination of band edges which require much more precision. Table 1-3. Comparison of experimental with theoretical values of band gap bowing. In,-,Ga,As Linear interpolation Deviation from linearity (x = 0.5) VCA Site CPA MCPA Supercell Expt.

ZnSe,Te,-,

Egap

Egap

530-13Ox

3040-33Ox

-59 -79

+

- 66 - 60

-

-123, -133

-

10

5 90 - 49 -270

1.7 Systems with Lower Dimensionality

57

(a) MCPA

I

(b) Bond-Centered CPA

-12

-8

-4

0

4

-12

Energy (eV)

-6

0

6

Energy (eV)

Figure 1-34. Density of states in MCPA, bond-centered CPA, and the supercell approach for

Figure 1-35. Density of states in MCPA, bond-centered CPA, and the supercell approach for

1n0.5Ga0.5As.

ZnSe0.5Te0.5.


tures, then analyse the use of the envelope function approximation, especially the problem of the correct boundary conditions. We then treat the case of quantum wells and superlattices. We also discuss hydrogenic impurities and excitons. Finally we present results obtained for silicon quantum dots.

With the progress of growth techniques it is now possible to build semiconductor structures which have two-, one, or even zero-dimensional character. Such structures are not only important for semiconductor devices but they also exhibit a lot of interesting physical properties. We shall shortly describe some basis aspects of these systems. We begin with their qualitative fea-

1.7.1 Qualitative Features Molecular beam epitaxy and other growth techniques have allowed to deposit

58


For a system confined in 3 - d directions and free in the other d directions, &k is the dispersion relation in d dimensions (with k the corresponding wave vector), while E,, are the levels resulting- from the confinement. We can rewrite n ( & )as (1 -1 63) [ d&’6(&- En - &’) a(&’ - E k ) n(&) =

semiconductors layer by layer. The simplest system one can obtain in this way is the quantum well, corresponding, for example, to a layer of GaAs within bulk GaAlAs (Fig. 1-36). The GaAlAs gap is larger than the GaAs gap, and the conduction and valence band offsets are such that one gets attractive potentials both for electrons and holes. Assuming that one can treat both electrons and holes in an effective mass approximation, this leads to a standard quantum mechanical one-dimensional problem except for the free particle motion in directions parallel to the layer. More recently i t has become possible to grow quantum boxes in which the particles are confined in all directions. Let us concentrate on the electrons, for instance, and discuss the general features of the density of states in such systems. By definition this is given by:

n

The quantity

&E’

- & k ) defines theden-

k

’ ) the ddimensional free sity of states v ~ ( E of electron gas (only defined for &’>O). This allows us to express n ( E ) in the form

(1-164) For a free electron gas in d dimensions, it is well known that Y ( E ) = ed’*-’ so that we get, quite generally, (1 -165)

This is plotted in Fig. 1-37 for bulk systems, quantum wells, wires, and dots, each showing remarkably distinct behaviors. It is also possible to produce gradual transitions between 2D and 3D behavior, for instance. For this, one can increase the width of the layers in which case the E,, will tend to form a pseudocontinuum as discussed in Sec. 1.1. One can also build a superlattice with equidistant quantum wells. When these are close enough, the states E, broaden into 1D bands, the steps in the density of states of Fig. 1-37 become smoother and n ( E ) tends to 3D behavior. All these cas-

n Figure 1-36. Conduction and valence band square well potentials for a layer of GaAs in Ga,-,AI,As.

E

k

E

I Figure 1-37.

I

Schematic densities of states for: a) bulk material, b) a quantum well, c) a quantum wire, d) a quantum dot.

59


es represent nice examples of general considerations about band structure formation.

1.7.2 The Envelope Function Approximation

1 df continuous. These reduce m * dz to the normal conditions for materials with the same effective mass. They also have the advantage of leading to the conservation of the current probability across the interface, which is a constraint that must necessarily be fulfilled. An illustration of the problem raised when choosing the boundary conditions is provided by a 1D tight binding s band model with diagonal energies U , or UB and nearest neighbors interactions PA or h depending on the material A or B, the coupling between A and B being taken as /3 (see Fig. 1-38). Writing the wave function C, qn, where qnis the “s” orbital of atom

tinuous,

The electronic structure problem of these structures is more complex than for bulk materials, especially when space-charge effects are included, in which cases the potential has not only discontinuities but also macroscopic curvature (e.g., n-p heterojunctions). Fortunately, in regions where the potential is slowly varying and where the carriers are close to the band extrema, it is possible to simplify the treatment by using the effective mass approximation discussed previously in Sec. 1.1.4.For a single band extremum the wave function takes the form of Eq. (1-32), Le., it is the product of the Bloch function at the extremum times a slowly varying envelope function F(r) which is the solution of the effective mass equation, Eq. ( 1 -30). This equation can be solved in regions where the potential has no discontinuity, but then one has to match the solutions from both sides of one discontinuity. Let us illustrate this at a simple interface where there is a conduction band discontinuity along the z direction; at z = 0 between materials A on the left and B on the right. Assuming isotropic effective masses mi and mg, the envelope function takes the form ~ ( r=f(z> ) eikll.rll

functions which must satisfy the standard boundary conditions and not their envelope part. It is, however, customary to use the following boundary conditions: f ( z ) con-

(1- 166)

where 11 stands for quantities parallel to the interface plane. The matching problem only concerns fA(z) and fB(z). For such a potential in a vacuum, the matching would require continuity of the wave function and of its derivative. However, herefA(z) andfB(z) are not the true wave functions but envelope functions in materials with different bulk properties. Normally it is the total wave

--

n

n and the C,,are unknown coefficients, one gets the following set of equations: ( E - UA) cn = P A (cn-l+ Cn+])n - 1 ( E - u.4)CO = P A c-1 + /3c1 (1 -1 67) ( E - UEJ c1 = PB c2 + PCO (ECn= (Cn-i+ Cn+l) n 2 For bulk states of the infinite materials A or B, the Bloch theorem tells us that the coefficients C, behave as Coexp(ikna) where a is the lattice parameter and the energy bands are given by U A , B + 2 p ~ , ~ c O S ( k a ) . The extrema of these bands occur at k = O or k = d a and the Bloch function at these

- - -* P A -2

usP A us P -1

0

% 1

PB

y 2

PB

*--3

Figure 1-38. The A . B interface for a linear tight binding chain with the intraatomic and nearest neighbors’ parameters.

60


extrema is such that C, = Co (+ l),. To make use of the effective mass approximation for the interface problem we must consider energies falling near an extremum for both materials A and B; this extremum corresponds to uA? 2 P A and UB & 2k,and the nature of the extremum is not necessarily the same in both materials. In this case we write cAn CBn

= cAO = CBI

"+"

sign, the value of the functionsf,,, at the interface, and (2) for the "-" sign, the value of a where f' is the derivative. Consideringfandf' as the components of a column vector, one can then formally write: (1-172)

where T is a 2 x 2 transfer matrix given by:

fAn

EG-lfBn

(1) for the

(1-168)

= 1 , gB = + 1 , the sign dependwhere ing on the nature of the extremum. In this case fArl and fB,I are expected to be weakly dependent on n and one can express them as the value taken by the continuous envelope functions fA(:) or fB(z) at I= nu. These envelope functions are solutions of the bulk equations, Le., the first and fourth Eq. ( 1 - 167), given by (1-169)

(1-173) This expression of the boundary conditions shows that the simple boundary conditions used in most calculations are not generally obeyed. In general, one should replace them by the transfer matrix formulation Eq. ( 1- 172) which has been generalized in Ando et al. (1989); Ando and Akera (1989) to realistic band structures. It is interesting to discuss the situations for which the transfer matrix Eq. (1-173) leads to the usual boundary conditions. For instance continuity of the envelope function (fA=fB) requires that TI, = 1 and T I , = 0. This last &B condition implies that P 2 - - PAh.The &A

signs of the different coupling parameters are obviously the same; the requires that EBIEA= 1, Le., that one is dealing with extrema of the same nature on both sides. If this is realized, the T matrix becomes:

(1-174) As the functionsf,,,(z) are slowly varying, the quantities f [ ~ A , B ( u >+ f A , B ( o ) I give:


The sign of can be absorbed as a change in phase of the functionf,(z). In this case, continuity of the wave function is ensured only if h/pA=1 corresponding to equality of the effective masses for the two materials. If this is so, continuity of the first derivative is automatically fulfilled. This is the only situation for which the usual continuity equations automatically apply. This seems to hold true for GaAIAdGaAs systems as shown in Ando et al. (1989); Ando and Akera (1989).

1.7.3 Applications of the Envelope Function Approximation The simplest application of the envelope function approximation corresponds to the isolated quantum well for single isotropic band extrema of the same nature in the two materials. This occurs for the conduction band of GaAs-AI,Ga, -,As systems in which the minimum is of symmetry in both materials when 0 c x c 0.45. Furthermore, the effective masses are of the same order of magnitude in the two materials. As we have seen, the well and barriers are in the GaAs and GaAlAs parts respectively. In such a situation the normal boundary conditions should apply and one has thus to solve a simple square well problem, the electron mass being replaced by the effective mass m*.The superlattice case is treated as a Kronig-Penney-type model and leads to a similar broadening of the quantum well levels into bands. The corresponding valence band problem under the same conditions is not as simple. As we have seen in the impurity case, it is necessary to solve a set of coupled differential equations which can more or less be simplified after some approximations as discussed in Altarelli (1986); Bastard (1988). Another interesting problem concerns the behavior of hydrogenic impurities in

r

61

quantum wells. Again the basic case concerns the isotropic band minimum with effective mass m*. In three dimensions the m*e4 = 5.8 meV binding energy is R* = 2A2 E? in GaAs. The same problem in two dimensions gives a binding energy equal to 4 R * . This exact value is of interest in understanding the trends as a function of the quantum well thickness. The first calculation of this problem was performed variationally (Bastard, 1981) for a quantum well bounded by two infinite barriers. The variational wave function was written as ~

(1-175) where z is the direction perpendicular to the layer, zi the impurity position, @thein plane distance from the impurity, and xl(z) the state of the ground quantum well sub-band. As expected, the resulting ground state binding energy increases from its 3D value as the thickness decreases. This method runs into difficulties in the small thickness limit where effective mass theory should not apply in the z-direction. In this case, it is more appropriate to perform a strict 2D application of effective mass theory where the variational function is taken as a product of the exact Bloch function at the bottom of the lowest sub-band times e x p - a e (Priester et al., 1984). This gives the 2D limit exactly by construction and is valid as long as the binding energy is smaller than the intersub-band separation. Finally, the case of acceptor impurities is extremely complex and we do not discuss it here (references can be found in Altarelli, 1986; Bastard, 1988). It is interesting to look at the exciton states and the corresponding optical absorption in systems with reduced dimensionality. Let us first consider a quantum well and

62


the simple effective mass equation, Eq. (1 - 1 12), for the exciton envelope function. The problem is complicated by the fact that there is confinement in the z-direction. There can still be free motion of the center of mass in the directions x and y parallel to the layer. The ground state obviously corresponds to no such motion and two variational wave functions for this state have been sought (Bastard, 1988):

where = ( x , - xh )’ + (ye - yh )’, N , and N 2 are normalization constants, A is the variational parameter and xle(z,) and XI,,(&) are the lowest states in each quantum well in the absence of electron hole interaction. With these trial wave functions one gets the result that the binding energy increases when the width L of the well decreases to reach a limiting value of four times the bulk one, as for hydrogenic impurities. It is interesting to calculate the oscillator strength by using the same method as in Sec. 1.3.2. Using for xIand x2 a sin ( k z ) form, one readily obtains from Eq. (1- 1 17)

For strong confinement, iltends to its 2D limiting value a / 2 so that this ratio becomes equal to ( 1 - 180)

where L is of the order of the interatomic spacing. This ratio can thus become, for GaAs, as large as 300 showing that the 2D confined exciton has much more oscillator strength that the bulk one. This is confirmed by experimental observations (Dingle et al., 1974). Let us finally try to extend this to the quantum box. We then consider a spherical box with infinite potential at the boundary r = R. We only consider the limit of strong confinement where the kinetic energy terms dominate the electron-hole attraction (they scale respectively as 1/R2and l/R), The latter can thus be included in the first order perturbation theory. At the lowest order, the envelope function for the lowest excitonic state takes the form

re

rh

with k=rc/R. One can now evaluate IM0,exc,12 for this OD case from the general expression Eq. (1 -1 17). This gives 2

IM0,exc.12

=IM~I

(1-182)

(1-178)

Thus the strength of this exciton relative to the bulk one for the same volume of material given here by V = $ n R 3 is equal to

Thus the relative strength of optical absorption between the quantum well and bulk exciton is, for the same volume V = S L of material, given by

(1-183)

(1-179)

which is the result of Kayanuma (1988). Again the bulk exciton radius is a -- 100 A while in the limit of very strong confinement R = 10 p\ or less. This means that the

63


enhancement is still larger than in quantum wells.

1.7.4 Silicon Quantum Dots It is now well accepted that the luminescent properties of porous silicon (Psi) (Canham, 1990) are at least partly due to quantum confinement effects. Indeed this complex material is thought to be composed of nanocrystallites between which electron and holes can tunnel. This means that direct optical transitions which are forbidden in the bulk crystalline material become allowed due to breakdown of the k selection rule. We thus review here the theoretical results for idealized silicon crystallites and compare them with experimental data. We mainly concentrate on the predictions for the band gap versus size (blue shift) and the fine structure of the levels related to exchange splitting. We also consider the case of amorphous Psi. We finally review other predictions for the existence of selftrapped excitons, the radiative and nonradiative recombination rates, and finally the dielectric properties of such crystallites. Let us start with ideal spherical crystallites. Their electronic structure has been calculated using different techniques: the effective mass approximation (EMA), empirical tight binding (ETB), empirical pseudopotential (EPS) and finally the “ab initio” local density approximation (LDA). As discussed before, EMA is a continuous approximation which reduces the crystallite to a finite spherical potential well. All other methods require specification of the atomic structure, which is taken as a spherical silicon atoms cluster with its surface dangling bonds saturated by hydrogen atoms. Figure 1-39 gives what are believed to be the best predictions for the energy gap versus size. This gap is defined as the difference in energy between the LUMO (low-

bulk

I

I

1/6

1/3

I

I

112

1

lld (nm-I)

Figure 1-39. Energy gap versus confinement parameter I/d for hydrogen terminated clusters: ETB results (Proot et at., 1992; Delerue et al., 1993) (continuous line), EPS (Wang et al., 1994) (X), LDA results of (Delley et al., 1995) (03)and Hirao et al., 1993) (+).

est unoccupied molecular orbital) and HOMO (highest occupied molecular orbital). The strikingly good, agreement between the ETB (Proot et al., 1992; Delerue et al., 1993), the EPS (Wang et al., 1994), and LDA predictions can be noticed (Delley et al., 1995; Hirao, 1993). (Note that the LDA gap values include a rigid shift of 0.6 eV, since this method underestimates the bulk band gap by this amount.) The reason why these three quite different methods give identical results is related to the fact that: i) they produce a bulk band structure of similar quality over an energy range of 20 eV, and ii) they correctly treat the boundary conditions at the cluster surface (Delerue et al., 1996). This is not the case for poorer ETB fits like the sp3s* (Hill et al., 1995) which underestimates the gap while the EMA leads to an overestimation. Figure 1-40 gives the comparison between theory and optical data. For this the calculated gap has to be corrected by the

64


2.2

-

5 2.0 v

x

p 1.8 -

s

1.6

-

.:

-

1

PL

1.4 -

..-

(Calcott et al., 1993; Suernoto, 1993) of some samples. In particular, an onset of a few millielectronvolts is observed. In addition, the lifetime of the luminescence decreases when going from low (4 K) to higher temperatures ( 100 200 K) in parallel with an increase in the luminescence intensity (Calcott et al., 1993; Zheng et al., 1992; Vial et al., 1993). Both effects have been interpreted on the basis of a "two level" model, the lower state being a spin triplet, the upper one a singlet, and the corresponding splitting being due to excitonic exchange (Calcott et al., 1993; Zheng et al., 1992; Fishman et al., 1993). However, afull ETB calculation, including spin-orbit and the excitonic exchange, has shown that this level structure is not obtained for spherical crystallites (Martin et al., 1994). This is due to the strong degeneracies of near band-gap states for silicon. It is only when considering strongly asymmetric crystallites that some splitting and quenching occurs in such a way that the two level model becomes justified, in agreement with experiment. Figure 1-41 compares the predictions for the exchange splitting [corrected with respect

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Diameter (nm) Figure 1-40. Compilation (Lockwood. 1994), of optical bandgaps of silicon crystallites and porous silicon samples obtained from optical absorption (empty symbols) and luminescence (full symbols and also the more recent results (Ehbrecht et al., 1997)). The dashed and continuous lines represent the calculated values (Proot et al., 1992; Delerue et al., 1993) with and without the excitonic correction.

electron-hole attraction (Proot et al., 1992; Delerue et al., 1993). A long-standing puzzle is the large difference between optical absorption and previous luminescence energies (much larger than the expected Stokes shift of the order of 25 meV for a gap of 2.6 eV). As shown on Fig. 1-40, the optical absorption data practically coincide with the predictions, while the previous luminescence data compiled in Lockwood (1994) are much lower with a weaker dependence on size. However, one group (Ebrecht et al., 1997) has reported luminescence measurements which are now quite close to the theoretical values. Finally, quite recent combined theoretical and experimental studies conclude that the large Stokes shift can be attributed for oxidized samples to the existence of Si=O bonds at the interface (Wolkin et a]., 1999). Although porous silicon is a heterogeneous material on a microscopic scale, a fine structure clearly appears in the excitation spectrum of the visible luminescence at 2 K

50

1

t

+

+

x A

+

0

A

+

01

1.5

'

I

1.7

'

I

1.9

.

X 2 X O

'

I

'

2.1

1

2.3

"

2.5

'

2.7

Gap (eV)

Figure 1-41. Excitonic exchange splitting; experimental values: full symbols; calculated values (empty symbols) for different orientations of the asymmetric crystallites (Martin et al., 1994).

1.8 References

to those of Martin et al. (1994) by using unscreened exchange interactions] versus size, with the two sets of experimental values showing pretty good agreement. Complete interpretation of the experimental data for the temperature behavior would require detailed analysis of the population of more excited states. We briefly review here other calculations performed on silicon crystallites: Amorphous silicon clusters: Many authors have considered that such clusters are not likely to give rise to quantum confinement and a blue shift of the gap. ETB calculations show, on the contrary, that amorphous hydrogenated silicon clusters give practically the same blue-shift dependence on size as the crystallites (Allan et al., 1997). - Recombination rates: The radiative recombination rates have been calculated (Proot et al., 1993; Delerue et al., 1993) to be 1-10 m s-*. Nonradiative recombination via surface neutral dangling bonds was also shown (Proot et al., 1993; Delerue et al., 1993) to eliminate the luminescence of the corresponding crystallites, whereas charged dangling bonds will luminesce in the infrared, as observed. Finally, Auger recombination is fast, in the nanosecond range, as calculated in Delerue et al. (1995) and Mihalescu et al. (1995). - Self-trapped excitons: Some aspects of broadband luminescence might be ascribed to the possibility of forming selftrapped surface excitons [as calculated in Allan et al. ( 1 996)] by high energy photo excitation. - Dielectric properties: It has been demonstrated (Lannoo et al., 1995) that the effective dielectric constant of nanometersized crystallites is much smaller than the bulk value. -

65

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Hedin, L., Lundquist, S. (1969), Solid State Physics, 23, I . Henderson, B. (1984), Appl. Phys. Lett., 44,228. Hill, N. A., Whaley, K. B. (1995), Phys. Rev. Lett. 75, 1130. Hirao, M., Uda T., Murayama, Y. (1993), Mater. Res. SOC.Proc. 283, 425. Hohenberg, P., Kohn, W. (1964), Phys. Rev., 136 B, 864. Hollinger, G., Jugnet, Y., Tran Minh Duc (1977), Sol. State Comm., 22, 277. H u h , M. (1966), J. Phys. Chem. Sol., 27, 441. Hybertsen, M. S., Louie, S. G. (1985), Phys. Rev. Lett., 55, 1418. Hybertsen, M. S . , Louie, S. G. (1986), Phys. Rev. B 3 4 , 5390. Ibach, H., Rowe, J. E. (1974), Phys. Rev. B 1 0 , 710. Ibanez, A., Olivier-Fourcade, J., Jumas, J. C., Philippot, E., Maurin, M. (1986), Z. Anorg. A&. Chem. 6, 5401541, 106. Jackson, W. B. (1982), Solid State Comm., 44, 477. Johnson, N. M., Biegelsen, D. K., Moyer, M. D., Chang, S . T., Poindexter, E. H., Caplan, P. J. (1983), Appl. Phys. Lett., 43, 563. Johnson, N . M., Jackson, W. B., Moyer,M. D. (1985), Phys. Rev. B 3 1 , 1194. Kane, E. 0. (1956), J . Phys. Chem. Sol. I, 82. Kane, E. 0. (1957), J . Phys. Chem. Sol. 1, 249. Kane, E. 0. (1976), Phys. Rev. B13, 3478. Kayanuma, Y. (1988), Phys. Rev. B38, 9797. Kittel, C . (1963), Quantum Theory of Solids. New York: Wiley and Sons Inc. Kittel, C., Mitchell, A. H. (1954), Phys. Rev. 96, 1488. Knief, S . , Von Niessen, W., Koslowski, T. (1998), Phys. Rev. B58, 4459. Knox, R. S. (1963), Theory of Excitons, Sol. State Phys., Supplt. 5: Seitz, F. and Turnbull, D. (Eds.) New York: Academic Press. Kohn, W., Sham, L. J. (1965), Phys. Rev. 140 A, 1133. Lannoo, M. (1973), J. de Phys., 34, 869. Lannoo, M., Allan, G. (1978), Sol. Stare Comrn., 28, 733. Lannoo, M., Allan, G. (1982), Phys. Rev. B 2 5 , 4089. Lannoo, M., Decarpigny, J. N. (1973), Phys. Rev. B 8 , 5704. Lannoo, M., Schluter, M., Sham, L. J. (1985), Phys. Rev. B 3 2 , 3890. Lannoo, M., Delerue, C., Allan, G. (1995), Phys. Rev. Lett. 74, 3415. Leman, G., Friedel, J. (1962), J. Appl. Phys. Supplr. 33, 281. Lefebvre, I., Lannoo, M. (1997), Chem. Mater. 9, 2805. Lefebvre, I., Lannoo, M., Allan, G., Ibanez, A,, Fourcade, J., Jumas, J. C., Beaurepaire, E. (1987), Phys. Rev. Lett., 59, 2471. Lefebvre I., Lannoo, M., Allan, G., Martinage L. (1988), Phys. Rev. B 3 8 , 8593. Lefebvre, I., Lannoo, M., Olivier-Fourcade, J., Jumas, J. C. (1991),Phys. Rev. B 4 4 , 1004.

1.8 References

Lefebvre, I., Szymanski, M. A., Olivier-Fourcade, J., Jumas. J. C (1998), Phys. Rev. B58, 1896. Lempert. R. J., Hass, K. C., Ehrenreich, H. (1987), Phys. Rev. B 36, 1 1 11. Lockwood, D. J. (1994), Solidstate Commun. 92, 101. Louie, S . G. (1980), Phys. Rev. B22, 1933. Louie, S . G. (1996), Quantum theory of real materials, Chelikowsky, J. R. and Louie, S . G. (Eds.), Kluwer, Boston, p. 83. Luttinger, J. M., Kohn, W. (1955), Phys. Rev. 97, 869. Luttinger, J. M. (1956), Phys. Rev. 102, 1030. Maley, N., Beeman, D., Lannin, J. S . (1988), Phys. Rev. B38, 10611. Markov, M., Zotov, N. (1996), Phys. Rev. B55, 2938. Martin, E., Delerue, C., Allan, G., Lannoo, M. (1994), Phys. Rev. B50, 18258. Martinez, E., Yndurain, E (1981), Phys. Rev. B24, 5718. Mauger, A., Lannoo, M. (1977), Phys. Rev. B15, 2324. Mihalescu, I., Vial, J. C., Bsiesy, A., Muller, F., Romestain, R., Martin, E., Delerue, C., Lannoo, M., Allan, G. (1995), Phys. Rev. B51, 17605. Mikkelsen, J . C. Jr., Boyce, J. B. (1982), Phys. Rev. Lett., 49, 1412. Mikkelsen, J. C. Jr., Boyce, J. B. (1983), Phys. Rev. B28, 7130. Mikkelsen, J. C. Jr. (1984), Bull, Am. Phys. SOC.,29, 202. Motta, N. et al. (1985), Sol. Stare Comm., 53, 509. Northrup, J. E., Hybertsen, M. S . , Louie, S . G. (1991), Phys. Rev. Lett. 66, 500. Northrup, J. E. (1993), Phys. Rev. B47, 10032. Nozieres, P. (1964), Theory of Interacting Fermi Systems, Benjamin, New-York. O’Reilly, E. P., Robertson, J. (1983), Phys. Rev. B27, 3780. Pantelides, S . T. (1978), Rev. Mod. Phys., 50, 797. Pantelides, S . T. (1986), Phys. Rev. Left., 59, 688. Pantelides, S . T., Harrison, W. A. (1976), Phys. Rev. B13, 2667. Petit, J., Lannoo, M., Allan, G. (1986), Sol. Stare Cornm., 60, 861. Perdew, J., Levy, M. (1983), Phys. Rev. Lett., 51, 1884. Philippot, E. (1981), J. Solidstate Chem., 38, 26. Poindexter, E. H., Caplan, P. J. (1983), frog. Surf. Science, 14, 201. Priester, C., to be published. Priester, C., Allan, G., Lannoo, M. (1983), Phys. Rev. B 28, 7 194. Priester, C., Allan, G., Lannoo, M. (1984), Phys. Rev. B29, 3408. Proot, J. P., Delerue, C., Allan, G. (1992),Appl. Phys. Lett. 61, 1948. Resca, L., Resta, R. (1979), Sol. State Comm.,29,275. Resca, L. Resta, R. (1980), Phys. Rev. Lett., 44, 1340. Robertson, J. (1983), Advances in Phys., 32, p. 383. Rohlfing, M., Kriiger, P., Pollmann, J. (1993), Phys. Rev. B48, 17791.

67

Sawyer, J. F., Gillespie, R. J. (1987), Progress Inorg. Chem., 34, 65. New York: Intersciences Publi. J. Wiley and Sons. Schiff, L. I. (1955), Quantum Mechanics, 2nd ed. New York: McGraw-Hill, p. 158. Schliiter, M. (1983), Proc. Enrico Fermi Summer School, Varenna. Schliiter, M., Joannopoulos, J. D., Cohen M. L. (1974), Phys. Rev. Lett., 89, 33. Sham, L. J., Rice, T. M. (1966), Phys. Rev. 144, 708. Sham, L. J., Schliiter, M. (1983), Phys. Rev. Lett., 51, 1888. Sham, L. J., Schliiter, M. (1985), Phys. Rev. B32, 3883. Shevchik, N. J., Tejeda, J., Cardona, M., Langer, D. W. (1973), Sol. State Comm., 12, 1285. Slater, J. C. (1 960), Quantum Theory ofAtomic Strucrure, Vol. 1. New York: McGraw-Hill. Slater, J. C., Koster, G. J. (1954),Phys. Rev. 94, 1498. Street, R. A,, Zesch, J., Thomson, M. J. (1983),Appl. Phys. Lett., 43, 672. Strinati, G. (1984), Phys. Rev. B29, 5718. Stutzmann, M., Biegelsen, D. K. (1989), Phys. Rev. B40, 9834. Suemoto, T., Tanaka, K., Nakajima, A , , Itakura, T. (1993), Phys. Rev. Lett. 70, 3659. Talwar, D. N., Ting, C. S . (1982), Phys. Rev. B25, 2660. Vial, J. C., Bsiesy, A., Fishman, G., Gaspard, F., HCrino, R., Ligeon, M., Muller, F., Romestain, R., Macfarlane, R. M. (1993), Microcrystalline Semiconductors-Materials Science and Devices, Fauchet, P. M., Tsai, C. C., Canham, L. T., Shimizu, I., Aoyagi, Y. (Eds.) MRS Symposia Proceedings no 283 (Materials Research Society, Pittsburgh), p. 241. Vogl, P., Hjalmarson, H. P., Dow, J. D. (1983), J. Phys. Chem. Sol., 44, 365. Wang, L. W., Zunger, A. (1994), J. Phys. Chem. 98, 2158. Weaire, D. ( I 981), in: Fundamental Physics of Amorphous Semiconductors, Springer Series in Solid State Science, Yonezawa, F. (Ed.). New York: Springer Verlag, p. 155. Weaire, D., Thorpe, M. E (1971), Phys. Rev. B 4 , 2508. Wolkin, M. V., Jorne, J., Fauchet, P., Allan, G., Delerue, C. (1999), Phys. Rev. Lett. 82, 197. Wooten, F., Winer, K., Weaire, D. (1985), Phys. Rev. Lett. 54, 1392. Wooten, E, Weaire, D. (1987), Solid State Phys. 40, 1. Zheng, X. L., Wang, W., Chen, H. C. (1992), Appl. Phys. Lett. 60, 986. Ziman, J. M. (1979), Models ofDisorder. Cambridge University Press. Zunger, A., Jaffe, J. E. (1983), Phys. Rev. Lett., 51, 662.

2 Optical Properties and Charge Transport

. .

R G Ulbrich IV. Physikalisches Institut. Georg.August.Universitat. Gottingen. Germany

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.6 2.6.1 2.6.2 2.6.3 2.7 2.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Optical Properties of Bulk Crystals . . . . . . . . . . . . . . . . . . . . . 71 Band Structure and Electron-Hole Pair Excitations . . . . . . . . . . . . . . 72 Dielectric Polarization and Response Function . . . . . . . . . . . . . . . . 74 Correlated Electron-Hole Pairs: Excitons . . . . . . . . . . . . . . . . . . .78 Single-Particle versus Pair Excitations . . . . . . . . . . . . . . . . . . . . 79 Exciton-Photon Coupling: Polaritons . . . . . . . . . . . . . . . . . . . . . 80 Interband Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . . 83 Direct versus Indirect Transitions . . . . . . . . . . . . . . . . . . . . . . . 83 Radiative Recombination Rate . . . . . . . . . . . . . . . . . . . . . . . . 84 Angular Dependence of Interband Transitions . . . . . . . . . . . . . . . . 85 Fundamental Gap Spectra at Low Excitation . . . . . . . . . . . . . . . . 85 Exciton Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Systems with Restricted Dimensionality . . . . . . . . . . . . . . . . . . 89 Electronic Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 92 Quantum Wells. -Wires and -Dots . . . . . . . . . . . . . . . . . . . . . . Optics of Quantum Confined Systems . . . . . . . . . . . . . . . . . . . . 94 Microcavities and Photonic Band Gaps . . . . . . . . . . . . . . . . . . . 94 Charge Transport and Scattering Processes . . . . . . . . . . . . . . . . 95 Momentum and Energy Relaxation of Carriers . . . . . . . . . . . . . . . 98 Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Carrier-Carrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Coherent Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Spin Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 107 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Optics and High Field Transport . . . . . . . . . . . . . . . . 109 110 Nonlinear Dielectric Response . . . . . . . . . . . . . . . . . . . . . . . . Carrier distributions far from equilibrium . . . . . . . . . . . . . . . . . . 111 Carrier Dynamics on Ultrashort Time Scales . . . . . . . . . . . . . . . . 112 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

70


2.1 Introduction Interest in charge transport and optical properties of semiconductors dates back half a century. The experiments of Shockley and co-workers upon transport of photo-injected minority carriers in germanium and silicon clarified for the first time the three basic phenomena which are common to electrical transport in semiconductors (Shockley, 1950): (i) carrier energies exceeding kT, (ii) various types and energy dependence of carrier-lattice coupling, (iii) controllable camer densities.

They pointed at the fundamental importance of electronic band structure for charge transport and optical properties of semiconductors, and how both differ from metals. Their work on Ge and Si soon initiated the exploration of III-V- and II-VI compounds. Scientific progress in this field accelerated considerably when the enormous potential and far-reaching practical applications of semiconductor devices became evident in the late 1950s and early 1960s. The last three decades have brought tremendous diversification to semiconductor physics. Starting with the work of Esaki and coworkers in 1970, earlier visionary ideas (Kroemer, 1985) on carrier confinement in and transport through heterostructures with geometrically designed and chemically “engineered” band structures were developed into real devices (for a recent review, see Esaki, 1993). In these years, steady progress in understanding semiconductors in the broader context of solid state physics was accompanied by an unparalleled development of sophisticated growth and processing techniques for semiconducting materials. The refinement of evaporation techniques, notably layer-by-layer deposition of atoms under ultra-high vacuum con-

ditions, as well as photon-, electron- and ion-beam mediated lithographic procedures led ultimately to the fabrication of manmade nanostructures on the atomic scale. Somewhat less spectacular, but equally important, the fabrication of ultrapure and highly perfect bulk crystal material was improved steadily, so that at present two prototype semiconductors - silicon and gallium arsenide - are by far the best-understood crystalline solids. The fabrication of increasingly complex devices with restricted dimensionality on the nanoscale, as well as the availability of well-defined semiconductor bulk materials, have produced new and unforeseen physical phenomena. Quantum size effects in geometrically confined systems play the key role: Their generic wealth is manifest in the unique properties of so-called quantum wells and superlattices, quantum wires, and quantum dots. Carrier ensembles in such systems, and their peculiar transport properties in external electric and magnetic fields, have now become topics of great practical interest. Spectacular new findings have come about in the context of localization and charge transport in semiconductors: the integral quantum Hall effect in 2-d confined carrier systems (von Klitzing et al., 1980; von Klitzing, 1986); strong Coulomb correlation of carrier motion in such systems leading to the fractional quantum Hall effect (Tsui et al., 1982; Stormer, 1984); quantized conductance in laterally confined quasi- 1d configurations (Beenacker and van Houten, 1990), the Coulomb blockade in fully confined systems with few electrons (Kastner, 1992), and Aharonov-Bohm-type interference phenomena in mesoscalic ring structures (Yacoby et al., 1995). During the past two decades, the main focus of scientific interest has shifted from quasi-classical, i.e. particle-like transport phenomena in bulk semiconductors to their coherent,

2.2 Optical Properties of Bulk Crystals

i.e. wave-like and phase-sensitive quantum mechanical counterparts in ensembles with few electrons. Closely related is the growing interest in optical properties of structured semiconductors and their widespread applications in optoelectronic devices. The renaissance of the seemingly mature field of classical optics, which had started with the invention of the laser in 1961, continued with key contributions from semiconductor physics: injection lasers, quantum dot lasers, microcavity devices, efficient photon detectors and emitters in the near infrared, optical nonlinearities and bistability, ultrafast optical switching and coherent pulse generation and propagation are examples. Expanding experimental abilities in synchronization and phase control of ultrashort laser light pulses have caused also here a shift in scientific interest: from studies of incoherent carrier population and energy relaxation effects to the coherent control of polarization waves, with descriptions of scattering beyond perturbative schemes, in an effort to understand the dynamics of optically coupled pairs on the microscopic level. Electrical transport and optical properties of semiconductors have been treated accordingly in many review articles and text books. The aim of the present article is to give a comprehensive survey of some of the basic concepts, to illustrate them in specific situations, and to discuss some of the key experimental results. The level of presentation should help the unfamiliar reader to become aware of the state-of-the-art in the field and assist him or her in evaluating the contemporary original literature. Elementary coverage of electronic band structure and optical transitions, the effects of carrier confinement and carrier scattering, and electrical transport will be given to some extent. This article does not cover the aspects of magneto-optics and magneto-

71

transport, but does contain some selected material on nonlinear transport and nonlinear optics in semiconductors. For further information on these topics, the reader is referred to the monographs and the review articles cited at the end of this chapter.

2.2 Optical Properties of Bulk Crystals The optical properties of bulk crystalline semiconductors in the visible part of the spectrum are closely connected with their electronic band structure. All optical spectra reflect essentially the unperturbed energy spectrum E,@) of filled electron states below, and empty ones above the Fermi Here n energy, with wavefunctions qn,k(r). is the band index, k the electron wavevector. To a certain degree, details of optical spectra in real crystals are modified by the presence of impurities and structural defects. At finite temperatures T>O, the agitation of thermal lattice vibrations broadens and shifts the optical spectra. Calculations of electronic band structures - at any level of sophistication - use input parameters which are derived from spectroscopic data. In retrospect, practically all important physical quantities related to semiconductor band structures were extracted from empirical spectra of absorption, emission and reflectivity. The most relevant parameters are fundamental and higher band gaps, the magnitude of momentum matrix elements, symmetries of wavefunctions at special points in the Brillouin zone, and joint density-of-state spectra of relevant bands. A common starting point for the discussion of electronic properties of crystalline solids is the spatially periodic crystal potential Vc(r):one assumes a priori that each crystal electron moves in the field of all

72


nuclei and all other electrons, and that all electrons see effectively the same potential. More precisely, the potential V c ( r ) is supposed to be of Hartree-Fock type, Le. a selfconsistent mean field (Anderson, 1963; Sutton, 1993). The use of V , ( r ) leads to the concept of the “one-electron” approximation: it is taken for granted that all electronic properties of the solid can be described in terms of products of mutually independent one-electron wave functions. Despite the large valence electron density of approximately ne = 3 x 1 023cm-3 in typical semiconductors, this concept works surprisingly well and is a good basis for all further treatments of semiconductor properties. Rigorous treatments of the many-electron ground state problem and justifications for the independent-particle description have been systematically developed during the past decades in the framework of densityfunctional theory (Hohenberg and Kohn, 1964: Dreizler and Gross, 1990). Ground state energies and corresponding charge density distributions of all relevant semiconductors have been calculated and found to agree very well with experiment (Payne et al., 1992). Excited states, and hence optical properties, are less well described in the density functional formalism. Though dynamical screening of an excited electronhole pair is difficult to treat, ab-initio calculations of gap energies and optical spectra in good agreement with experiment begin to emerge (Pulci et al., 1997; Aryasetiawan and Gunnarson, 1998; Rohlfing and Louie, 1999).

2.2.1 Band Structure and Electron-Hole Pair Excitations In this paragraph the connection between single-electron band structures E, (k) and fundamental optical spectra of semiconductors is described on an elementary level.

Excitonic effects and screening effects connected with high e-h pair densities will be discussed further in Ch. 2.3. Filled valence bands, empty conduction bands, and a band gap of width E@kT between them define the ground state of the N-electron system “semiconductor” with energy E,. Since all N electrons may conduct a current, in principle, the terms valence- and conduction band are not meaningful in this context. What counts is the distribution of occupied and empty states in each energy band E, ( k ) with band index n. An electron in an otherwise empty band, and a hole in a filled band are both charge “carriers”. They are the electronic excitations which constitute the response of the semiconductor to external electric fields, i.e. charge transport and optical properties. An excited state E* of the N-electron system may be created by taking first an electron out of a valence band state, and putting it into an empty conduction band state (see Fig. 2-1). This procedure defines the creation of an electron-hole (e-h) pair. electron

N aN+1

eA

N

* N-1

hole

Figure 2-1. An excited state of the N-electron system (left). The energy-wave vector diagram (right) shows valence band hole and conduction band elec(h) tron with their kinetic energies and Ekin . The total energy E* of the excitation includes the potential energy E, (see text).

€EL


The total energy change AE connected with this process is the sum of four terms: (i) potential energy of the electron with respect to V c ( r ) .It is called “gap energy” E, and corresponds to the change in band index n (the analog of the atomic orbital quantum number) ; (ii) kinetic energy E$, of the initial electron state, resp. the final hole state; (iii) kinetic energy EgA of the final electron state, resp. the initial hole state; (iv) the interaction energy Vi,, between the excited electron and the remaining hole. At this level of the single-particlescheme, any rearrangement of the valence electron density caused by the excited e-h pair is neglected. We may therefore write

For wide enough bands and sufficiently small kinetic energies of the carriers the socalled effective-mass-approximation (EMA) applies (Cohen, Chelikowski, 1988; Yu and Cardona, 1996; Chow and Koch, 1999), and the kinetic energies are simply given by

and are counted from the band edges E, and E,. The coefficients m: and m$ represent the effective masses of electrons and holes (see also Ch.1.14). Electrons and holes of given wave vectors k, and k h are distinguishable particles if their band index n is different. They can be described by fermion creation and annihilation operators, which act on the “Fermi seas” of filled valence and empty conduction band states (Haug and Koch, 1994). If we want to go further than counting numbers, we have to study wavefunctions, and the discussion has to be more specific

73

(Harrison, 1980; Yu and Cardona, 1996). The prototype semiconductors which we have in mind are crystalline solids with tetrahedral coordination. This class includes the group IV element Ge, the group 111-V compound GaAs, and the 11-VI compound ZnSe. These three prototypes have their chemical homologues, which appear in the columns of the periodic table of elements: the monatomic C, Si, Ge family, the diatomic semiconductors InP, InAs, InSb, and CdS, CdSe, ... etc. It turns out that these materials, which all crystallize in either diamond-, zinkblendeor wurtzite-lattice structures, have many basic features in common: their non-bonding conduction band states at and around E, have the symmetry of atomic L=O or “s-like” orbitals, derived from the symmetric linear combination of sp3 hybrids on neighboring sites, with even parity in the diamond case, and twofold degeneracy (for the electron spin). Their effective mass is usually small compared to m,, i.e. m ~ / m , c1. The valence band edge at energy E, has eigen-functions similar to “p-like” orbitals. They are derived from the anti-symmetric linear combination of sp3 hybrids, with odd parity in diamond structure. Its sixfold degeneracy is split by the spin-orbit interaction such that the j = 3 / 2 terms lie usually above the j = 1 / 2 split-off level (Kane, 1966). In the diatomic zinkblende and wurtzite lattice structures the potential Vc ( r ) and hence the valence electron density ( r ) is somewhat different on the cation and the anion site. This asymmetry causes characteristic linear terms in the E,&) relation at k = O (Kane, 1957; Cardona et a]., 1986; Yu and Cardona, 1996). Away from the zone center k = O the valence band extremum is split by the k . p interaction into heavy and light hole bands, because of the spatial anisotropy of the sp3 hybrid wave functions which form the

74


in each lattice cell: “s“ Electrons i

I

P

I

__ ,/-

,

d

/

-

, =21’

I m f= * -2

& I d l -

- 1

“p,” Heavy Holes

___ti

.

A

d

~

t

-

8

-

i

’,

“px+ipY” Light Holes

-

J = 3? , i

,

m , = +12

J

*z, k, Figure 2-2. Cell-penodic wave functions u k ( r )in a cubic lattice for navecector k 1 : Only the spatial part I S shown, the spin parts la) are chosen to be aligned in :-direction (schematic)

Y”,~ ( r )= v - ” ~ e i k ’ ruk(r)

(2-3)

Common basis functions for the symmetry classification of the cell-periodic part of the wavefunction, uk(r),are either the atomic orbitals “s”, “p”, “d” etc. in Cartesian coordinates, or, including spin, the eigenfunctions Yj,,, belonging to the total angular momentum operators J = L + S , with j = 3/2 or 1/2. The overlap integrals between wave functions in neighboring cells for a given wave vector direction (Le. chosen quantization axis) depend on the component mj of J in that direction. This is the physical origin of the difference between heavy and light hole mass; it can be inferred qualitatively from Fig. 2-2. Lucid discussions of crystal symmetry and band degeneracy can be found in Tinkham (1964), Bassani et al. (1 975), and Yu and Cardona (1996).

2.2.2 Dielectric Polarization and Response Function tetrahedrally coordinated bonds (Harrison, 1980; Sutton, 1993). For the same reason, the effective masses at k = O and beyond are anisotropic, in general. With increasing k , the effective mass parameters m $ usually become larger. In the wurtzite structure (with a different bond length along the c-axis), the valence bands are nondegenerate at k = 0. The basic shapes of cell-periodic wave functions and their nearest neighbor overlap are shown qualitatively in Fig. 2-2. A simple 2-dimensional quadratic lattice representing fourfold coordination in a monatomic crystal has been chosen for illustration. The proper eigenstates of both electrons and holes in the periodic crystal potential V , ( r )are usually described by Bloch-wavefunctions. They extend over the entire crystal volume, with equal probability density

Optical studies of semiconductors probe their dielectric polarizability. At sufficiently high frequencies the response of any solid is dominated by electronic transitions. When static or low frequency electric fields act on a diatomic, (Le. polar) semiconductor, there is an additional effect: the two sublattices in zinkblende (or wurtzite) structure will move with respect to each other and give a lattice contribution to the induced polarization. In the following we will restrict ourselves to excitation frequencies that are high in comparison to the vibrational frequencies (L),,b of the lattice. The natural frequency oo of a simple mass-spring system falls in the infrared part of the spectrum for all common values of lattice force constants D and atomic masses M in semiconductors (see, e.g. Ashcroft and Mermin, 1976). D is directly related to


75

E = O , p=O

the inverse bulk modulus r a n d the interatomic distance (bond length) d through (2-4) We find that the coo is in the range 0.32 x 1014 ... 2.4 x I O l 4 rad/s when we go from the heaviest 11-VI compound CdTe to the lightest group IV crystal diamond (C). In the visible part of the spectrum ( w = 3 x 1015rad/s at 2 ev), o+wvi, is ~ U I filled and we may safely neglect the contribution of lattice polarizability, because the nuclei cannot follow the driving field. We will now discuss the dielectric properties of semiconductors as a function of incident photon energy fiw, or light frequency. The low frequency limit, or “static” case, implies frequencies which are small compared to characteristic electronic excitation energies. Lattice effects will not be considered further. Let us consider as the driving force a spatially uniform electric field E , * exp (-i w t ) which interacts with all N electrons in the crystal. The induced polarization p of electronic charge in the directed bonds and (to a much lesser degree) in the inner shells of the atoms will, in general, cause a counteracting and non-uniform electric field in the solid and hence reduce the “effective” field at the atomic sites. The importance of such local field corrections depends on the degree of ionicity and the localization of the bond charges. The reader is referred to the article by Lannoo in this volume for a thorough discussion (see also Resta, 1983; Baldereschi et al., 1993). For simplicity we neglect here the local field effect altogether and keep in mind that the assumption of a uniform electric field implies weak polarizability of electronic wavefunctions (see Fig. 2-3). The electronic polarization density P within the crystal is the sum of all induced

E

.

Figure 2-3. Charge density distribution around nuclei in periodic crystal with no external electric field (upper) and induced dipole momentsp = X Q E caused by external electric field (lower).

polarizations pi per volume:

P=

v-’cpi

(2-5)

i

The electric field due to the crystal potential V , ( r )close to the nucleus is lo8V/cm. Provided that the external E , is much smaller than this, it is reasonable to assume that the induced P is proportional to E,. This basic assumption defines the regime of “linear optics” in solids. For ordinary incoherent light sources, where power densities

-

S = - E10

ClE01*

(2-6) 2 do not exceed a couple of kilowatts per cm2, the condition is perfectly well fulfilled:

E,= 19 V/cm

for S = 1 W/cm2

(2-7)

Only laser light sources offer the possibility of entering the nonlinear regime. Focused and short-pulsed lasers are now routinely used to study the nonlinear optical response of solids (see Sec. 2.6 below). The proportionality constant between P and E is the dielectric polarizability x, or

76


phase component to x2 :

“response function”, defined by

P = e,

~ ( 0 E )

(2-8)

in the frequency domain, with sinusoidally varying fields

E = E , . e-’W‘

(2-9)

Here it is understood that only the real part of E represents the actual electric field. Summing over frequencies and including appropriate phases in Eq. (2-9) allows to describe any time dependence of E ( t ) without loss of generality. % ( w ) will, in general, be a complex function of w

x ( w )=xl( w )+ i

x2 ( w )

(2- 10)

x,

with an in-phase “dispersive” term and an out-of-phase “absorptive” term x2. We can extract the dielectric response from the electronic band structure as follows: the driving field E is treated as a small perturbation H ‘ to the Hamilton operator H,.The latter contains V , ( r )and stands for the unperturbed crystal. The vector potential is chosen such that

E = - c - I - =~ - AAio at

(2-1 1)

c

for the plane wave of Eq. (2-9). We add the field momentum e . A / c to the momentum operator p in H,. To lowest order in A the coupling term between a single electron and the light field H = - ep . AA = - -

eh

E.Vk (2-1 2) mo 0 Here m, is the free electron mass. The next term is proportional to A 2 and is neglected here (Bassani and Pastori Paravicini, 1975). The incorporation of Eq. (2-12) into timedependent perturbation theory is straightforward. With the aid of Fermi’s “Golden Rule” one obtains the induced current density in the crystal and relates its out-of-

(2-13)

(Ec( k ) - E, ( k ) - A O )

V is the volume appropriate for normalization of V,,Jr) , the sum includes all wavevector pairs (e,k ; v , k ) which fulfil the condition that the initial state is full and the final state is empty (with a plus sign), or vice versa (with a minus sign). The driving field is chosen along x-direction, and P is taken to be parallel to E . Equations (2-8)(2-13) are the simplest and qualitatively correct semiclassical approach to the linear optical response of semiconductors:

(i) absorption of light is mediated by the linear coupling between E and single electron momentap in Eq. (2-12); (ii) the sum in Eq. (2-13) contains transitions from occupied valence to unoccupied conduction band states (“upward transitions”) and their time-reversed counterparts (“downward transitions”). With the appropriate signs, Eq. (2-13) describes both fundamental optical processes of induced absorption and emission.

0 0

A

0

,.

m0

V

Figure 2-4. Light absorption from the crystal ground state, x2>0 (left), and light emission from excited state, x r < O (right). Both elementary processes are contained in the dielectric polarizability (Eq. 2- 13, see text)

77


The extension towards a full quantum mechanical description is conceptually simple: the vector potential A is quantized and the elementary process of photon absorption (emission) occurs only if the two coupled states (c, k ; v,k ) differ by Ao in energy. Since the photon momentum Ac q lo5 cm-’ in the visible part of the spectrum is small on the scale k,,, = n / a O 6 x 1O7 cm-’ of electron quasi-momenta A k , one speaks of “vertical” transitions in k-space. The matrix element of Eq. (2-13) may be rewritten by using the relation

-

-

which is valid for localized electron orbitals (Sutton, 1993). Equation (2-14) connects the operator for the dipole moment, e - 2, with ~ ( o and ) , allows to substantiate the intuitive interpretation of Fig. 2-3: the process of optical absorption may be seen as the time-proportional admixture of “s-like” wave function amplitude to the initial “plike” valence band wave function mediated by the dipole moment operator 2. The superposition of both eigen-states produces an oscillatory motion of the bond charges with amplitude x out of their equilibrium positions x = 0. Inspection of Eqs. (2-5) and (2-14) shows that the expectation value of x corresponding to one optically coupled e-h pair per site in the crystal is

It is convenient to introduce the momentum matrix element g~(Lawaetz, 1971) (2-16) According to Eq. (2-15), we find that for a typical semiconductor with @ = 30 eV,

E,= 1.5 eV, the induced dipole moment is 1x nm per V/cm and elementary charge. Equation (2-13) reveals that all optical transitions are directly connected with momentum matrix elements -just the same quantities which also determine the band gaps and band curvatures near band extrema in the framework of k * p perturbation theory. This connection leads to a number of useful relations (so-called sum rules) which help to determine mass parameters and g-factors from optical spectra, and vice versa (Kane, 1966; Lawaetz, 1971; Harrison, 1980; Yu and Cardona, 1996). Up to this point we have only discussed the dielectric polarization for resonant excitation with light, i.e. Aw2EG. It follows from very general arguments, which are based only on locality, causality and linearand x2 (0) are ity of the system, that x1(0) closely connected with each other. One relation is (Kronig, 1926; Kramers, 1927; Stem, 1963; Cardona, 1969):

-

It gives an explicit expression for

x1(0) 2

2~ e

c

1 l(k,cIa/axIk,v)l

x1(@=-7 m0 k,k’ o k k ’

v

2 2 okk’-(L)

(2-18)

x

(w) describes the The real part of response of all electrons weighted with a resonance denominator. The form of the dielectric susceptibility resembles very closely the behavior of an ensemble of massive charges on springs which are characterized by individual eigen-frequencies o’= (Eck- E,,)& and are driven by an external force field with frequency o.This analogy with forced harmonic oscillations has been recognized very early - in fact long before quantum mechanics -in the work of Thomson, Drude

78


and Lorentz. They interpreted the empirical results of their time, which are now described quantum-mechanically by Eqs. (2-13) and (2-1 8), in terms of electro-mechanical oscillators (Jackson, 1975). For a given frequency w one must sum over all possible resonances of Eq. (2-18), i.e. all e-h-pair states with zero total momentum. The occupation of initial and final states is taken into account by summing only over filled valence and empty conduction band states (or vice versa, with opposite sign). For the consistent description of matter in Maxwell’s equations the complex dielectric function is defined E ( ~ ) = E o ( I+%(W))

(2- 19)

It describes the dielectric response of a semiconductor and is the quantum mechanical version of the response of a linear system to harmonic time-dependent perturbations (Kubo, 1957). An important extension of Eqs. (2-13)(2-1 8) can be made in order to include the possibility of partially filled (resp. empty) states k , and k , . Each term of the k-sum is multiplied with scalar distribution functionsf,, f h obeying 0 one ~ tries to describe the relative motion of e and h in terms of plane wave states originating from the unperturbed conduction and valence bands. Solutions of the wave equation for this problem of screened exciton motion have been extensively discussed (Zimmermannet al., 1978; Haug, 1985; Haug, 1988). We skip here a thorough discussion of details and sketch only plausible aspects of the problem. Suppose the relative e-h motion of a single exciton is described by the envelope

Eih’,

2.4 Systems with Restricted Dimensionality

function Qls

=

v-fexp

(-2)

(2-38)

The expansions of this wavefunction into plane wave states I k , ) and I kh) have Lorentzian shapes, are centered around k , = k h = 0 and extend to roughly ai-’ in k-space (Landau, Lifschitz, 1966). If we consider the available density of states in this wavevector range, we can estimate that approximately a density of n, ai-3 excitons can be created in a given crystal volume before single particle phase space occupations reach f,-fh 1. A further increase in e-h pair density requires more kinetic energy, because of the Pauli principle, and increases the Fermi energies EF’ and EAh’. The interplay between the increase of kinetic energy and lowering of the screened potential of the e-h pair motion controls the behavior of the system as a function of density. There is a relatively smooth transition from individual excitons at low density into a plasma of highly correlated e-h pairs around n,, and finally into a dense metallic plasma with density much higher than n,, and less pronounced long range correlation effects (Rice 1977; Haug, 1985).

-

-

2.4. Systems with Restricted Dimensionality In the past two decades, much effort in engineering of semiconductor materials has been directed towards controlled carrier confinement in one ore more dimensions (Esaki, 1993). Growth techniques such as molecular beam epitaxy have made it possible to fabricate semiconductor structures down to nm scales. The almost arbitrary indepth control of material composition and impurity doping, and the utilization of selforganized growth have been key elements.

a9

Not only fundamental gap energies and band-offsets, but also band structures in general have been purposely designed for electrical transport and optical properties. The ability to synthesize nanoscale structures, such as 2-d wells and superlattices, l-d wires, and O-d dots has in turn spawned new classes of semiconductor devices. Two classes of mesoscalic structures can be distinguished: down-scaled conventional device geometries (like e.g. a field-effect transistor) and structures made to exploit quantum effects. The former class is usually operated in regimes of temperature and electric field strengths, for which the electron coherence length is smaller than the smallest device dimension: charge carriers are still treated as classical particles. In quantum confined structures, on the other hand, the de-Broglie wavelengths AB of electrons close to EF are fundamental to the phenomena in focus. Here, one or more structural dimensions are usually made comparable to AB, and are typically much smaller than the phase-breaking length &. It is a measure for the mean distance over which the phase spread of the propagating electron wave due to fluctuating scatterers is of the order one. The inelastic mean-free path L E ,which is the distance electrons move on the average without loosing energy, is determined by the electron-phonon interaction (see Sec. 2.5 below) and is of the order of 1 .. . 100 nm at room temperature, but can reach tens of pm at low T. Mesoscalic devices exhibit properties directly connected with the wavemechanical properties of electrons, even though the overall structure may still be many orders of magnitude larger than the underlying atomic scale of the solid, characterized by its lattice parameter ao. A semiconductor structure with restricted dimensions resembles a low-loss electron waveguide more than a bulk conductor.

90


-

2d growth

2d - MQW

I d - wire

Od dot

-

Figure 2-10. Semiconductor structures of restricted dimensionality: starting with planar crystal growth techniques (liquid phase, gas phase, or molecular resp. atomic deposition), followed by lateral structuring with lithographic (wet etch, dry etch, ion beam) techniques. devices with typical length scales L, and quasi 2-d, l-d and O-d electronic behavior can be made.

But the guiding is defined by a potential Vconf(r),unlike the conducting walls of an ordinary electromagnetic waveguide. The essential point is that the long-range Coulomb part of Vconf(r)can be easily controlled from outside, e.g. an external charge reservoir. Such devices offer the unique possibility of exploiting quantum-mechanical interference for information handling and storing purposes. At ambient temperature T , an electron in any specific quantum state may pick up the energy AE of order kT from thermal fluctuations. If A E is less than the energy difference between adjacent discrete states (confined in three, two, or just one spatial dimension) one speaks of a quasi-0, 1 , or 2-dimensional system. To characterize the system further, one defines three length scales L,, L,, L, (see Fig. 2-10). If one, two, or three of these quantities tend towards zero, the electron’s spatial degrees

of freedom are said to be fully confined to 2, 1, or 0 dimensions. If the lengths are reduced such that only the weaker AE criterion is met, i.e. AEconf>kT, then the prefix “quasi”-n-dimensional is more appropriate. Lithographically defined patterns on the 10 nm to 100 nm scale have produced a large variety of such electron wave-guides and resonators (Wilson et al., 1993). The terms quantum-“well”, quantum“-wire” and “-dot” are now established shortforms for structures with typical lengths L, between the lattice constant a, of the atomic constituents, and the inverse of a relevant electron (hole) wavenumber k : a,,< L,< k-’

(2-39)

Here k represents the magnitude of the thermal wavevector of a carrier distribution, or the Fermi wavevector kF of a denser ensemble of carriers which is confined in the structure (Capasso, 1993). Fig. 2-10 shows the geometries of quasi 2-d, l-d and O-d structures made on planar substrates by atomic deposition with indepth control of material composition and subsequent lithographic or cleavage techniques to laterally build the structure. Buried structures based on Stranskii-Krastanov growth have been developed (Woggon, 1997). In an effort to obtain ideal, i.e. atomically smooth interfaces on larger scales, Tshaped wire structures were fabricated by overgrowing in-situ freshly cleaved surfaces (Wegscheider et al., 1994, 1997). However, substitutional disorder still affects the confinement potential. The assembly of any crystalline heterostructure, via doping in a pn-junction, or via heteroepitaxy in the advanced systems, has always to cope with a principal problem: different materials have, in general, different lattice constants. In practice, the use of ternary alloys allows to “lattice-match”


materials with different electronic gap energies and valence band edge energies. But the strain fields which are induced in composite structures can as well be exploited for specific applications (Bauer and Richter, 1996; Pryor et al., 1997)

91

r

2.4.1 Electronic Confinement The fundamental effect of confining a particle within a potential well has been known since the early days of quantum mechanics. Any deviation of the crystal potential V , ( r )(see Sec. 2.2.1) from translational symmetry, caused by chemical composition and/or structural disorder, will modify the spectrum of electron energies. There are two basic types of interaction (see Fig. 2-1 1): (i) electron and hole change their energies in covariant form, (ii) electrons and holes change their energies individually because of local chemical bonding. The first mechanism is connected with the long-range Coulombic attraction (repulsion) of the carriers by a donor (acceptor) impurity. More general, any localized charge distribution, like an impurity point charge, a line charge, a charged layer, or a 3-d space charge cloud, causes this kind of covariant change of E, and E,. To transcribe the concept of extended states k , with global eigen-energies E (k,) (see Sec.2.2) to a local desription in r-space, we consider an electron at rest, define for the ideal periodic case (i.e. L, -) its total energy E at and around r as E, , and add to this the confinement energy AEconffor the actual L,. An infinitely deep well which confines just one degree of freedom, gives

-

(2-40)

Figure 2-11. The two types of carrier confinement: (i) covariant change of electron and hole energy caused by long-range Coulomb attraction (like e.g. donor impurity) and (ii) individual energy change due to local variation in chemical binding.

For m $ = 0.1 mo the confinement energy is AE = 10 meV for L , = 5 nm. The hole case is more complicated because of the 6-fold band degeneracy at k , = 0. Strong mixing of j=3/2 and j = 1/2 states occurs there and leads, in general, to a manifold of valence bands with highly anisotropic and non-parabolic E ( k ) dispersion relations (Weisbuch and Vinter, 1990). The second type of interaction is the local chemical bond effect: the cell-to-cell variation of V,(r) in H, leads to states which cannot, in principle, be classified as Bloch states (see Fig. 2-1 1). There are changes of E, and E , with chemical composition, that can be either covariant or contravariant, and hence cause edge variations on relatively small length scales: the gap energy EG changes “locally” with chemical composition. Ab-initio calculations for ideal planar interfaces have been made and showed that band offsets are not uniquely determined by the bulk properties of the constituents, but that interface-specific phenomena also exist, especially for heterovalent IV-III-V systems (Baldereschi et al., 1993).

92


In randomly substituted 111-V alloys an approximate k-representation can be justified (Wang et al., 1998). Pseudopotential calculations of single-particle energies and wavefunctions for edge states in disordered 111-V alloys have been made (Mader and Zunger, 1994). In the more phenomenological effective mass approximation, the breaking of translational symmetry at, e.g. a heterojunction, is simply described by the spatial variation of the effective mass parameters, rn;(r)and ntE(r), over short distances. The effective mass operator is incorporated into the kinetic energy according to (Ridley, 1997; Foreman, 1998)

In the following we discuss the various manifestations of carrier confinement in structures of 2 , 1 and 0-d.

2.4.2 Quantum Wells, -Wires and -Dots

In addition to the more or less “abrupt” discontinuities of band structures at an hetero-interface, the transfer of charge between the constituents causes so-called bandbending. This effect plays a central role for the overall performance of a heterostructure, a superlattice or a quantum well (see Fig. 2-1 2). To describe this effect, one starts with a single-particle trial wave-function, (r) calculates the charge distribution throughout the system, solves the Poisson equation to find the potential distribution, and adds the latter as an external quantity

V,, ( r )=

lr-r’i

dr’

(2-42)

to the crystal potential Vc(r).This is repeated until self-consistency is reached. In planar geometry it is straightforward and

E A x

/’

well

substrate

Figure 2-12. Potential landscape of quasi-2d structure (quantum well) and 1-d quantum wire. Band bending caused by charge transfer from the constituents (bulk, substrate) controls the equilibrium situation in a self-consistent way.

gives global potentials Vtot(r)as shown in Fig. 2-12 (Bastard, 1988; Grahn, 1995). With suitable doping of the bulk (substrate) material and/or application of an external electrical field, a metallic quasi-2d electron gas can be established in the region of quantum confinement. The most important characteristic of such a 2-d electron gas is that the mobile carriers are supplied by the remote bulk material, so that there are no corresponding ionized charges to scatter electrons, as would be the case in a doped bulk semiconductor. The key characteristics of quantum wells made from 111-V compounds - almost ideal interfaces, and small effective electron mass -have resulted in electron mobilities more than two orders of magnitude larger than those of bulk Si. Nanostructures made of GaAsGaAlAs are at present the main tool for studying 2-d, 1-d and 0-d systems and their ballistic transport and optical properties.

93


The fabrication of quasi-ld structures (quantum wires) has been explored with lithographic techniques. Edge roughness and the effect of strong localization have hampered the clear demonstration of 1-d transport properties in wires with large aspect ratios. Despite much efforts, clear experimental evidence for full lateral con-

number of electrons, and also exhibit a discrete spectrum of energy levels. The dots may be connected to external reservoirs. In two-terminal configurations with weak coupling established via tunneling gaps (see Fig. 2-13), charge transport through the dot, and its capacitance with respect to the environment vary strongly when the charge Q, characterized by the electron number N,is changed by one single electron, QN+ QNkl = Q N f q e . The energy E needed to add or subtract an electron depends on the external potential V,,,, (see Fig. 2-13), and the electrostatic self-energy of the dot, expressed by its capacitance C: n2

E = Q . Vg,te

+ X.-

(2-43) 2c It can be directly measured by monitoring, as a function of the potential difference between the reservoirs, the current through the dot. Capacitance C scales linearly with spatial dimensions, e.g. for a sphere with = 4 JC eo R lo-’’ F radius R one has Csphere at R = 10 nm, and a sufficiently small system will, according to Eq. 2-43, exhibit quantum effects even at room temperature. The quantization of charge causes pronounced peaks in the current which are peri-

-

Edd*

.,

N-I

O M =N e

I

.’ N --;q \

I

AV

I

-.Y- - - - \_e E , (left) and in the case of charge transport induced by a slowly varying electric field with hW<EG.

carriers. The domain of “electrical transport” covers the frequency range 0 e A o4EG, that is, dc to high frequencies of -Terahertz. The regime of “optical excitation” is characterized by Am>&, that is ac fields with frequencies high enough to induce interband transitions. The ac conductivity (I(0) is directly related to the dielectric response function

o ( w ) = i Eo w [ 1 - & ( 0 ) ]

(2-48)

and describes the response of “free” resp. “bound” charges in a semiconductor to an external perturbation, that is electric current density j in transport, and dielectric polarization currents dPldt in optics. The real part of a(w)contains, as discussed in Sec. 2.2, the dispersive (in-phase) components o f j and P,the imaginary part all dissipative (out-of-phase) processes. Energy-conserving, i.e. elastic as well as inelastic carrier scattering mechanisms both contribute to (I. Optical excitation of electron-hole pairs with narrow-band, i.e. more or less monoenergetic laser light sources with A o-E, s kT creates ab initio non-equilibrium distributions with large deviations sffromfo, the equilibrium Fermi function at tempera-

98


ture T. More precisely, we can havef(k)s

fo(k,T) even in the regime of linear optics with arbitrary small external fields E 4 E , (see Sec. 2.2.2). Genuine electrical transport in a homogeneous sample, on the other hand, induces smoothly varying distributionsf, at least for those common scattering mechanisms which may be treated in relaxation time approximation (see below) and obey the smallness criterion 6ffi,,, ( k ) + f ( k ) . Both phenomena are insofar complementary; the combination of optical methods with traditional wire-bound techniques of electrical transport has emerged as a powerful analytic tool in the field of highspeed devices and circuitry testing (Auston et al., 1984; Fattinger and Grischkowsky, 1989; Nuss et al., 1998). Optical spectroscopy with its inherent energy selectivity (see Fig. 2-15) is able to extract very detailed information on specific scattering mechanisms in semiconductors. In fact, recent progress in time-resolved spectroscopy (see Sec. 2.6) has allowed qualitatively new insights and understanding of electron and phonon transport on a microscopic level (Wehner et al., 1998) .

2.5.1 Momentum and Energy Relaxation of Carriers Let us investigate in some detail the basic processes which relax the initial momentum and energy of a charge carrier. We consider a single electron or hole in an otherwise empty band, and assume that all other bands are either filled or empty. All single-particle excitations of the crystal - electrons, phonons, plasmons, excitons, etc. - may couple to the carrier under consideration, provided that momentum and energy conservation are fulfilled. If the kinetic energy Ekinof the carrier (measured from the nearest band extremum) satisfies €kin

/m*

(2-58)

The brackets (. . .) denote averaging in the sense of Eq. (2-56). The magnitude of the

low-field drift mobility ,u is a measure for the average strength of the scattering which relaxes the deviationf(k) of the actual distribution from the equilibrium distribution f , ( k ) .The action of the driving field E on the distribution function in k-space and at a given temperature T I , together with the “restoring force” represented by momentum scattering, A k, is shown schematically in Figs. 2-17 a,b. The same system at higher temperature T2 has, in general, a different S f ( k ) , resp. p, which “probes” larger k-values, because of the higher kinetic energies involved. It is straightforward to show that the temperature dependence of p can be expressed as a power law

-

p ( T ) T”

(2-59)

where v stands for the exponent in the energy dependence of the scattering time t= t(E)-€’ (Seeger, 1999). The second factor in Eq. (2-58) is the density of free carriers in the semiconductor. Doping with shallow impurities of one kind - donors or acceptors - populates at sufficiently high Tpreferably one band with one type of mobile charge carrier, electrons or holes. The majority carrier concentration in a strongly compensated n-type semiconductor is (Blakemore, 1974)

n (r)

-

(NA-ND)

(2-60)

. [ 1+ ( N A / N ~,) exp ( E D / ~ T ) - ’ where E, stands for the donor binding energy, and N A , ND are the concentrations of impurities, /3- 1, and NC-4x10l7 (T/300)3’2cm-3 in a typical case (GaAs). At high enough temperatures n ( r )- N D - N A . Each ionized impurity is a localized charge with a long-range static electric field around it. The ensemble of randomly distributed ions causes a static random electric field throughout the crystal which has pro-

2.5 Charge Transport and Scattering Processes

found consequences for charge transport: it scatters the mobile carriers very efficiently, especially at low temperatures. The mechanism of ionized impurity scattering is closely related to carrier-carrier scattering (see below) and gives a mobility pion-T3’2 (Seeger. 1999). The effective total mobility of carriers which are subject to several different scattering mechanisms can be obtained approximately by adding the reciprocal mobilities for each individual process, provided their z(E> dependence is identical (Ashcroft, 1976). This procedure, known as Matthiessen’s rule, is valid when quantum-mechanical interference effects between the scattering mechanisms do not play a role. It is of great practical interest to quote for a given bulk crystal its combined total mobility curve p ( T ) . This function is characteristic for each semiconductor material and type of carrier ( e , h), and is the basis for all device applications (Hess, 1980; Sze, 1985; Seeger, 1999). The presence of ionized impurities lowers in general the mobility, especially at low temperatures, and modifies the intrinsic p( T ) relations. The latter quantities are tabulated for all common semiconductors (Adachi, 1985; Landolt-Bornstein, 1989). Maximum values occur typically in the temperature range 20.. . l o 0 K, and vary from low 100 cm2/Vs in p-type materials to high lo6 cm2Ns in n-type crystals with small effective masses, like the 111-V compound GaAs. Undoped semiconductor crystals have thermally excited e-h pairs of density

no= N , exp (- Eg/2 k T )

(2-61)

where N , has a value of 8 ~ 1 0 cm-3 ’ ~ at m* =mo,and T= 300 K. For i-GaAs we find no lo6 cmp3 at room temperature. The net current flow in the ambipolar case is composed of two counter-propagating

-

101

particle currents: a hole current and an electron current. Application of Matthiessen’s rule would yield the ambipolar mobility

ptot=pe phI(pe+ph)

(2-62)

but this overestimates ptot considerably because of the strong relative momentum relaxation mediated through the attractive e-h Coulomb scattering (Auston and Shank, 1974; Kuhn and Mahler, 1989). A genuine disadvantage of homogeneous doping in the volume is the reduced mobility, especially at low temperatures, T e 100 K. It has been circumvented by so-called “modulation doping” (Dingle et al., 1979; Stormer, 1984), that is spatial separation of impurity centers from the mobile carriers. The application of remote-doping techniques has steadily increased the practical mobility limit in n- and p-type GaAs-GalxAlAsx heterojunctions and quantum well structures from io5 ... io7 cm2Ns (Pfeiffer et a]., 1989; Umansky et al., 1997). At this point it is useful to extract numbers from Eqs. (2-58 ... 60). A typical ntype bulk semiconductor, like e.g. GaAs, with relatively small electron effective mass m* and high mobility, has for the dop10l6, 10l8cm-3 mobiling levels n= ities as shown in Fig. 2-18. At room temperature, T = 300 K, we find A= lo4 cm2Ns, and a mean scattering time t- 3 x s or a corresponding mean free path of L-0.1 p. There is a trade-off between high mobility values and their practical realization in devices. It has to do with the onset of nonlinearities. As we discussed in the context of the transport equation, a large mean free path, i.e. long scattering time, causes a much stronger dependence of the mean carrier energy on driving field. “Hot carrier” effects combined with a drastic lowering of the mobility occur in such high-mobility materials at relatively low field strengths.

-

102

2 Optical Properties and Charge Tran sport

approach is correct if (i) the wavelength 2 d q is much larger than the interatomic distance, so that the crystal can be treated as a continuous elastic medium, and (ii) the carrier kinetic energy is small compared with the total energy band width. The deformation-potential mechanism occurs in all solids: any system of electrons and nuclei responds to enforced volume changes 6 V with changes 6 E of its total energy. The energy shift can be expressed as (Bardeen and Shockley, 1950; Shockley, 1950)

6 E = E,, div u ( r )

TEMPERATURE

(‘K)

Figure 2-18. Drift mobility as a function of temperature for bulk gallium arsenide under low-field conditions. Uniform doping levels are indicated.

Excess energies of E s kT have been found at low temperatures with fields 1 E I = 10 V/cm in a two-dimensional electron gas (Shah et al., 1983, 1985). 2.5.2 Phonon Scattering

This paragraph discusses the dominant carrier-phonon scattering mechanisms in some detail and gives an idea of the relative importance in different temperature ranges. Acoustic waves in a solid are accompanied by strain. The central idea of the deformation potential approach is that the electron-phonon coupling term H ’ of Eq. (2-46) is given approximately by the shift 6 E of the conduction band edge that is produced by a homogeneous strain in the crystal. It is assumed that the local strain which is set up by the phonon mode q in a given unit cell equals the strain for the q = O mode. The

(2-63)

and it is obvious that only longitudinal acoustic waves with volume changes div u - q u are accompanied by a non-vanishing deformation potential constant EDP. Transverse acoustic modes (shear waves) produce to first order in u no volume changes. The matrix element for deformation potential coupling is finally H’ = EDp 4 .U = A

1 1 +-+-

(2-64) (2@VAos)”* 1 2 2 Here is the density of the crystal, q the acoustic phonon mode with energy A o and the occupation number N , , and V the normalization volume. If we assume for a moment that carriers and phonons are in equilibrium with each other, the mean free path can be evaluated relatively easy: if the carriers are non-degenerate, i.e. obey a Boltzmann distribution, and if the phonon ensemble is also in an equilibrium distribution of the same temperature T , and if the electron-phonon scattering can be treated as quasi-elastic, i.e. €4kT, one finds for the mean free path of the electrons =

4

IN

L = 3 $I h4 8 us (4 m,*ED, kT)-’

(2-65)

with v, sound velocity, rn; carrier effective mass. L is independent of carrier energy. For


a typical semiconductor with m$=0.1 mo, EDp=10 eV, Q = 5 g / ~ m and - ~ v,= 3 x lo5c d s and T=300 K a value of L=2.3 pn is obtained. From Eq. (2-65) it follows that the scattering time is also the momentum relaxation time and is proportional to t- E-’

(2-66)

For excess energies E > kT, that is in the “hot electron” regime, the drift mobility hpwill therefore decrease with increasing electric field. In crystals with no inversion-center symmetry there is an additional acoustic phonon scattering mechanism caused by long-range electrostatic forces. An acoustic wave with wave-vector q displaces both sublattices by different amounts and induces a macroscopic dipole moment P,which in turn (see Sec. 2.2.2) produces an electric field (piezoelectric effect)

E = - (qpe/EO) div u

(2-67)

qpe is the piezo-electric constant (LandoltBornstein, 1989). The electrostatic energy density of this field represents a change in carrier energy H’=SE=(ee14/~Oq).divu =(ee14/&oq ) (2-68)

Jm

Here e I 4 is the relevant component of the piezo-electric tensor for zinkblende symmetry, which connects the induced dielectric polarization with the applied mechanical stress. The phases of both matrix elements, Eqs. (2-64,2-68) with respect to the lattice wave u (r,t ) are different and differ by n/2. This produces an interference between both scattering mechanism when they couple to the same phonon mode (Rode, 1975). It is interesting to note that for kinematic reasons there is a threshold energy for carriers to interact with acoustic phonons: the quadratic electron dispersion relation

103

-

E ( k ) k2 and the linear phonon dispersion relation o-q allow for single phonon emission only if the condition E>Emin=m*vi/2-5 peV

(2-69)

is fulfilled (Levinson, 1977). Optical phonon modes, i.e. lattice vibrations with large relative displacements in each unit cell, produce the same two types of interaction potentials. The short-range effect is called optical deformation potential scattering, and the long-range electrostatic part is the polar optical (or Frohlich) scattering. Since optical phonon energies are typically large, A wLo 20 .. . 50 meV, and the coupling is relatively strong, both processes provide a very efficient energy relaxation mechanism for carriers, provided Practitheir kinetic energy exceeds A oLo, cally all optical and transport experiments with monoenergetic carrier injection show the characteristic threshold for optical phonon emission, the solid state analogue of the Franck-Hertz experiment (Tsui, 1974; Heiblum and Fischetti, 1987; Becker et al., 1986; Sprinzak et al., 1998). The strong coupling of polar-optical phonons to electrons, the so-called polaron problem, requires more than first-order perturbation theory for an adequate treatment (Feynman, 1972; Devreese and Evrard, 1976).The matrix element of the interaction operator is

-

’ l l .-.l INq+-&-

(2-70) 47, 2 2 and favors small changes of carrier momentum. Since the energy transfer AE=A wLo is always high (A E>kT in all practical cases) a meaningful momentum relaxation time in the spirit of Eq. (2-55) cannot be defined at low temperatures. Both optical phonon coupling mechanisms with their

104


high threshold energy tend to dominate the total mobility curves p ( T ) if T is comparable with or larger than the Debye temperature, which is 400.. . 800 K for the common semiconductors. In some cases the coupling constants for deformation potential and polar scattering were determined empirically from a variety of different experiments and have been checked against each other. The range of experiments includes electrical transport with quantitative mobility curves p ( T ) (Rode, 1975), several methods of optical spectroscopy involving mechanical stress and isotope effects (Ramdas, 1995), tunneling spectroscopy (Tsui, 1974), free-carrier scattering, and finally inelastic light scattering (Raman- and Brillouin scattering) with absolute cross sections related to the e-phonon matrix elements (Grimsditch et al., 1979; Yu and Cardona, 1996). Despite this large number of independent experiments, there are still large ambiguities in the assignment of absolute values to certain coupling parameters. Even very basic quantities, like e.g. the individual deformation potential constants E,, for conduction and valence bands in common semiconductors, are only known with moderate precision. The example GaAs is discussed by Price (1985).

2.5.3 Carrier-Carrier Scattering The scattering of an electron by the change in V, ( r )due to the presence of other electrons is qualitatively different from phonon scattering. The latter process couples delocalized electronic states with lattice vibrations spread uniformly over the crystal volume, and the interaction between electrons and ions is relatively weak. The former process, that is the mutual interaction between mobile carriers - electrons, or holes, or both - depends on the inverse rel-

ative distance between both scatterers and may therefore be deeply inelastic for close encounters. As a consequence, it is not useful to describe the Coulomb scattering between carriers in a perturbative way. A standard approach along the lines of the tractable two-body problem is to treat it as a simple binary collision process in relative coordinates. At all times only one nearest neighbor is included. Higher order effects, like the presence of a third scatterer during the collision, as well as screening in the plasma of mobile carriers, and the coupling with collective electron motions (plasmons) are disregarded altogether. Nevertheless this approximation scheme has been used and reasonable agreement with experiments, at least in certain limiting cases, has been found (Appel, 1961; Platzman and Wolff, 1973). In the case of e-h scattering, the problem is closely connected with the spectrum of e-h-pair (or “exciton”) continuum states (see Sec. 2.2.3 above). For the sake of simplicity, transport theory starts usually ad hoc with plane wave states and tackles the problem in a perturbative way: the screened Coulomb interaction V(l re- r h 1 ) between two carriers in real space is taken to calculate the scattering rate in Born approximation. The rate of collisions between the two particles, with initial wave vectors k , and k , , and final wave vectors k ; and k ; , is then given by (Schiff, 1968)

(k;kS leV(rl?)lklk,)=

e’ exp(- lrl - r 2 1 / A

lrl - r2 I . e x p [ i ( k l r l+ k 2 r2)].drl. d r 2 4

(2-71)

EO

This expression is valid for two electrons in parabolic bands (i.e. in effective-mass-

2 . 5 Charge Transport and Scattering Processes

approximation) and for not too high energies because of the static screening used. The collision rate in Eq. (5-27) depends only on the relative separation between both scatterers. The usual two-body transformation of coordinates r l , r2 and k l , k2, k ; , k; into the center-of-mass reference frame simplifies this expression into

( k ; k;

I W12)Ik1k2)=

=-.e E

v

1 -k12

l2 +A-*

(2-72)

and shows now the complete equivalence of the initial problem with that of a particle of mass ,u (reduced mass) colliding with a fixed elementary charge, i.e. the familiar Rutherford scattering. From the wellknown scattering cross section for this configuration one can easily extract relative momentum and energy relaxation rates (Landau and Pomeranchuk, 1936). In the case of indistinguishable particles (two electrons, or two holes collide), the rates have to be slightly modified because of interference terms in the final state. The scattering rate following from Eq. (2-71) has been evaluated in detail for two limiting cases: a) one “hot” carrier interacts with a sea of “cold” equilibrium camers of given density n (Larkin, 1960). b) an initially monoenergetic distribution of carriers relaxes towards equilibrium (Hearn, 1980; Yoffa, 1980; Asche and Sarbej, 1984). For case (a) the energy relaxation rate is dE-n e4 . L ( E ,A) (2-73) dt 4 n ~ ~ 4 Here n is the carrier density, E the dielectric constant, m* the carrier effective mass, and E the instantaneous kinetic energy of the incident carrier, supposed to be much larger than the thermal energy kT. The function

105

L ( E , A) accounts for the details of screening and is only weakly energy and densitydependent. For low densities and in the range of practical interest ( n - 101 4 . . . 1017~ m - ~ ) one finds L 1. From Eq. (2-71) we obtain the relaxation rate z-l for the loss of relative momentum, i.e. the momentum loss in the center-of-mass coordinate system: -1 n e4 (2-74) z, = -

-

2 a n f ( & &0)2 @E312 Taking E = 10, E= 100 meV, m*=0.1 mo we find that z- 100 ps at n= 1017~ r n - ~ . There are only very few direct experimental checks on the validity of this description. Transport coefficients have been calculated with explicit inclusion of e-e scattering in the BTE (Wingreen et al., 1986). The stopping power of fast electrons in metals, and inelastic electron transmission through thin semiconductor crystals (e.g. in electron transmission microscopy) have been measured and compared to the rates discussed here (Elkomoss and Pape, 1982). Since the approach leading to Eq. (2-73) neglects all collective motions of carriers - plasmons and phonon/plasmon coupled modes - which may act altogether as very efficient momentum and energy relaxing scatterers, it will underestimate the total scattering rate (Petersen et Lyon, 1990). On the other hand the model takes into account only the nearest scatterer (Le. the deepest inelastic channel) and neglects the screening influence of all other electrons, which tend to reduce the scattering rate. Finally the pair correlation of electrons and holes is disregarded completely. All three effects enter the final rate in qualitatively different ways. More quantitative studies are needed to understand the process of carrier-carrier scattering in optically excited semiconductors, especially in the context of ultrafast carrier dynamics (see Sec. 2.6 below).

106


2.5.4 Coherent Phenomena Long-lived coherence of e-h pairs was discussed in Sec. 2.2.5 in the context of the polariton concept. With well-controlled laser pulses, the quantitative preparation of certain initial states is easily achieved in a semiconductor. During the laser pulse and when it is turned off, there is a regime in which the excited electronic states can evolve coherently. In principle, the wave function Y of an isolated system described by the Hamiltonian H , changes with time in a completely deterministic way:

-=--H()Y dY i dt A

(2-75)

With Eqs. (2-30, 2-31) considered as the full H,, and with polariton wave functions Y ( r ,t ) solving it, Eq. (2-75) describes the coherent evolution of Y ( r ,t ) under H,. Coherently prepared states do interfere, and they cause quantum beats with frequencies

Ao = -1( E , h

- Et))

(2-76)

where E, and E , are the eigen-energies of any pair of states involved in the process. The beats correspond to rapidly varying oscillations of classical variables, such as e.g. the macroscopic electric dipole moment P of Eq. (2-5). If, instead, we have a system that interacts with its environment, the full Hamiltonian has three terms: H , for the system by itself, Hintfor the interaction between both constituents, and He,, for the environment alone. The problem can be treated in different ways. Either the system and the environment are considered together, with a single wavefunction that evolves according to Eq. (2-75) with the full H,. Or, we continue to describe only the wavefunction of the system of interest, and we expect Eq. (2-75) not to be satisfied. Instead, the system

appears to evolve incoherently. One accounts for this information deficit by introducing phenomenological damping parameters, as e.g. the To in Eq. (2-24). stands for all interactions of the system with the environment (phonons, impurities, etc.), except those couplings, like e.g. the electron-photon coupling, which were included a priori in Ho and Hint.Often the wavefunction Y can be factorized into parts resembling different degrees of freedom of the system, like e.g. the spin and the spatial properties of an electron in an otherwise non-magnetic semiconductor.

r,

2.5.5 Spin Effects

Spin and optical coherence in semiconductors are closely related. Consider an electron in the conduction band of a semiconductor with an external applied magnetic field B,, having its spin aligned perpendicular to I, such that it is in a pure spin state. The spin wavefunction is given by ox+i oYand will evolve coherently by “precessing” (in classical language) in the plane IB,.An ensemble of electrons excited with circularly polarized light will relax with time constant T 2 , the so-called transverse spin relaxation time. Spatial inhomogenities will cause slightly different precession frequencies, so that the effective time constant T ; is smaller than T2 and the macroscopic magnetization

(2-77) Time-resolved measurements of the degree of polarization (linear, circular) of light emitted via interband transitions give direct information on the decoherence of the spin system caused by its environment (Zakharchenya, 1984; Oestreich et al., 1996). The method is known as H a d e effect


in atom spectroscopy. Typical relaxation times TZ 100 ps ... 1 ns have been found in undoped semiconductors, and exceedingly long T2 > 100 ns in n-doped GaAs.

-

2.5.6 Quantum Transport The concepts surveyed in the beginning of this section were originally developed in the classical context of the kinetic theory of gases (Chapman and Cowling, 1939). The central idea is the mean free path L together with the mean free time t between collisions of the atoms constituting the gas. Both quantities were simply reformulated for charge carriers in semiconductors. There are, however, limitations in the analogy. A plausible criterion for the applicability of the classical concept of t is the magnitude of the collision rate compared with the rate of energy change divided by the typical excitation energy AE (2-78) If the collision rate is much smaller than the latter quantity, the idea of individual collisions with well-defined kinematic properties is valid. In the other limit one expects qualitatively new behavior, so-called quantum transport (Haug and Jauho, 1996). It stands for conditions which require explicit consideration of the wave mechanical properties of the electrons and phonons involved in the scattering. Recent developments in miniaturization of MOS-FET devices with channel widths substantially below 1 pm and high field transport in layered structures gave some evidence for such intra-collisional effects and the necessity for modifications of the classical Boltzmann transport picture (Hu et al., 1989; Datta, 1997; Wacker et al., 1999). When the mean free path L and the confinement dimension LtyPbecome compar-

107

able, wave-mechanical size quantization becomes dominant and cannot be ignored. In this case interference terms may show up between the scattering amplitudes into different channels (Eq. 2-12), as well as finalstate interactions, and the concept of the collision term in the Boltzmann transport equation has to be checked with care. We mention here two eminent examples of quantum transport in confined carrier systems: recent experimental findings on quantized conductance G = gspine21h

(2-79)

in quasi-one-dimensional semiconductor structures (Beenacker and van Houten, 1988; Buettiker, 1988) and the quenching of the normal Hall-effect in thin-wire four-terminal configurations (Roukes et al., 1987 and 1990). They are striking consequences of the two basic effects which define the regime of quantum transport: quantization and interference. Like the energy, or the angular momentum, of any isolated system, the conductance of a small metallic system is also quantized. This fundamental effect was first observed experimentally in a 2-d confined electron gas (van Wees et a]., 1988). It lets the Fermi velocity vF of carriers and their density-of-states cancel exactly, so that the lateral confinement of an arbitrarily shaped constriction (see Fig. 2-19) leads to fully quantized 1-d dispersion relations. This result is independent of details of the electron group velocity dispersion in that direction. The conditions are such that L,ZL,, and that the constriction potential is sufficiently “smooth” to cause no reflection. In quasi-2d systems with a lateral confinement achieved by back gate electrodes, a series of elegant experiments has been performed by van Houten and others (van Houten, 1995). All aspects of electron “wave optics” were demonstrated: coherent point source, ballis-

108


tic flow, magnetic deflection, diffraction, focussing and defocussing, and two-slit interference. Closely related are the so-called conductance fluctuations in the current-voltage characteristics of narrow-channel MOSFET’s. At low enough temperatures the

A

T

,

B

€ ?

3 -

-2

--*+ kx

A€

Figure 2-19. The conductance of a small metallic system with a narrow constnction IS quantized at sufficiently low temperature T The inset shows the manifold of I-d electron dispersion relations, with quantized energy AE determined by the lateral confinement potential in the constriction region (schematic, see text)

mean free path for elastic scattering, L , , is long enough to establish phase-coherent transport of electrons close to E,. At helium temperature, La 1 pm in ordinary metals, and much larger, L,- 10 ... 100 pm in the 2-d electron gas of a high-quality semiconductor quantum well. The wavefunctions of stationary (current-carrying) states Y ( r ) exhibit B-field and and gate-voltage dependent variations, which are individual “fingerprints” of the spatial pattern of scatterers in the sample. Each individual device has its own characteristic I (V) dependence. In addition to these, one finds also “universal” conductance fluctuations, of magnitude Ag-e2/h, which are caused by weak localization of carriers in the current path and cause stochastic transmission steps as a function of B. For a lucid discussion, see the monographs by Beenacker and van Houten ( 1990) and Datta (1 997). On the other hand, a surprisingly wide class of experiments even in very small structures can be successfully described in terms of the simple wave-packet picture. This can be illustrated with the example of “hot carrier injection” through tunnel contacts into regions with a high potential gradient and a collecting electrode within a distance z - Lfree(Heiblum and Fischetti, 1987; Sprinzak et al., 1998).

-

$2) F

Figure 2-20. Ballistic transport of carrier wavepackets in a structured semiconductor with metal contacts and tunnel injection. The mean free path and the dominant energy relaxation processes can be extracted from geometry Lo and currentholtage characteristics of the device.

109

2.6 Nonlinear Optics and High Field Transport

During ballistic flight, see Fig. 2-20 below, the wave packets spread in space, and are eventually collected as a current in the back electrode. Measurements of the collection efficiency as a function of bias voltage Ueb can give direct information on the mean free path of carriers in the field region. Recent experiments in GaAs-GaAIAs hetero-junctions at room temperature have reported values of several tens of nm, in good agreement with the above mentioned mobility values. Closely related are recent experiments on ballistic transport of hot electrons through thin gold films via tunnel injection (Bell et al., 1990). A large part of the ongoing effort toward small devices is focussed at controlled ballistic transport in structures which can be fabricated and electrically controlled on a length scale L Lfree(Kelly and Weisbuch, 1986; Datta, 1997; Buks et al., 1998). The necessity to include quantum mechanical concepts is obvious. Phase-sensitive electronic devices may contribute to further progress in micro-electronics.

-

2.6. Nonlinear Optics and High Field Transport For sufficiently small driving fields E , observation tells us that transport and optical properties are independent of E. It has 0 such a “linear been shown that for E response” behavior follows from very general grounds (Kubo, 1957). If one considers, say, the ubiquitous scattering mechanisms discussed in Sec. 2.5, the range of “small” electric fields in semiconductors extends from 1 V/cm at helium temperature and low doping level to several kV/cm at room temperature and higher doping. When the driving field is no longer small and becomes a sizeable fraction of the internal crystal field of - lo8 V/cm, which char-

-

-

acterizes the ground state and cohesion properties of the system, we have to expect nonlinear response. At first sight it is surprising that metals usually do not show deviations from Ohm’s law, even under high field conditions. The large number of carriers involved in a typicm-3 cal metal - there are D (E) dEelectron states in the energy range dE = kT around the Fermi energy E, at room temperature - leads to an important consequence: the excess kinetic energy of the ensemble of carriers is very tightly coupled to the lattice. This prevents the electrons from getting “hot” even for extremely high current densities up to the order of 10” A/cm2, which corresponds to a driving field of 1O4 V/cm in good conductors, like Cu, Ag or Au (Heinrich and Jantsch, 1976; Schoenlein et al., 1987, 1988; Del Fatti et al., 1998). Semiconductors, on the other hand, are in general prepared with much smaller densities of active carriers. Doping with suitable impurities controls the carrier density in the range ne= 1015... 1 0 ’ ~~ m - ~i.e., approximately five to eight orders of magnitude less than in ordinary metals. The dissipation of momentum and energy has to be channeled through this “dilute” ensemble of carriers. At a given current density the mean carrier velocity - and hence carrier kinetic energy Ekin- is higher by the same factor. This explains why “hot carrier” phenomena are easily observed in semiconductors for modest electric fields ranging from several V/cm at low temperatures to kV/cm at room temperature (Shockley, 1950; Conwell, 1967; Hess, 1980; Sze, 1985; Shah et al., 1985; Weisbuch, 1986). If the same comparison is made for a fixed driving field strength we find that the phonon system acts as the relaxation “bottle neck”: in metals a sizeable fraction of the loz3cm-3 electrons feed energy into the lattice and produce Joule heating which - in

-

110


the stationary case - has to be transferred via phonon transport through the crystal boundaries into the external heat sink. Under such conditions phonon scattering controls the onset of nonlinearities. In a semiconductor the lattice vibrational modes have to take much less total power at the same electric field, and it is the nonlinearity within the carrier ensemble which controls the overall transport properties. However, at extremely low temperatures, in the milli-Kelvin range, hot electron effects may also be observed in metals (Roukes et al., 1990: Imry, 1997). For a more detailed discussion of nonlinear transport it is useful to expand the distribution functions f,(k,) and fh ( k h ) into a power series with the driving field as smallness parameter. The procedure is justified for the common scattering mechanisms which involve acoustic phonons. However, if only one highly inelastic process, like e.g. optical phonon scattering (see above Sec. 2.5.3) is present, the concept does not work (Peierls. 1955). An arbitrarily small electric field is now sufficient to drive the system into the nonlinear regime with carriers This threshold marks having Ekin> h qo. the onset of real phonon emission, i.e. the process of “polaron stripping” (Feynman, 1972). 2.6.1 Nonlinear Dielectric Response

The situation under resonant conditions, that is for driving frequencies at and above the fundamental gap, is more complicated and less well understood. An example is the edge spectrum of a direct gap semiconductor: it depends on the degree of excitation. The presence of electron-hole pairs or of phonons strongly influence the shape of x (0) at and around tZ ugap. The absorption of light, or heat, and also the presence of impurities and defects changes the energy spectrum E,@), alters the occupation of electronic states in the crystal, and modifies its optical properties, especially at and below E,. Some of the optically coupled states may be driven into high occupancy or even saturation and the concept of a series expansion of the susceptibility becomes questionable. Ab-initio treatments of this case have been made; they are closely connected with the above-mentioned high-field quantum transport phenomena (Haug, 1988; Combescot, 1988; Zimmermann, 1990; Haug and Jauho, 1996). The key point is the inclusion of wavemechanical ingredients, i.e. the phases of the electron and hole wavefunctions, into a description based on local phase-space distribution functions f,(k,) and fh ( k h ) (see Sec. 2.5). Wigner has introduced a natural generalization of these functions in the form of

The treatment of nonlinear optical properties under nonresonant conditions, that is in the transparent part of the optical spec. exp (-ik.R ) d3R (2-80) trum. starts with a similar expansion: the Here q ( r )is the wave function of the parinduced response P is written as a power ticle under consideration. The Wigner disseries of the driving field (Bloembergen, tribution function fw yields I 1c, ( r )I * when 1962). The description of nonlinear optical properties in terms of coefficients x‘”, ~ ‘ ~ 1 , integrated with respect to k , and its Fourier etc. has turned into an enormously successtransform 1 @ ( k ) I * when integrated with respect to r . It thus incorporates wave-like ful concept for practical applications (Levensohn, 1982; Mills, 1998: Chemla, 1999). properties of the particle (electron, hole),

111


but the straightforward interpretation offw as a local probability is restricted, because it is in general not non-negative (Wigner, 1932). The single-particle-approximation described in Sec. 2.2 works well for static screening and not too abrupt spatial variations in the perturbing potential qext.At the screening of densities around n, the e-h pair interaction leads to a screening length of the order of a;. In this intermediate density regime a more refined treatment is necessary. There are extensions of the Hartree-Fock self-consistent field equations which include screening from the beginning, and account for the response of the electron-hole plasma to a given charge carrier which is at rest, or is kept in motion by an external perturbation qextwith frequency o (Collet, 1989; Schafer, 1993). The canonical treatment due to Lindhard considers only those contributions of the induced screening charge density around the carrier which are linear in the total potential qtot= qext+ (Pin& The plasma is assumed to be in an equilibrium state described by Fermi functions f,( k ) ,fh ( k ) . Application of first-order time-dependent perturbation theory leads to the Lindhard dielectric function (Wooten, 1972)

-

e (w, q ) = 1+ -.

e2

q2 fo ( k ) - f o ( k + q )

(2-81)

Here o is the frequency of the external perturbation, q is its wavevector, and the sum extends over all occupied single-electron levels E ( k ) of the unperturbed system, and fo is the equilibrium distribution function at temperature T. The derivation assumes that the perturbation is stationary, and recovers for q 0 the Thomas-Fermi screening discussed in Sec. 2.2.

-

2.6.2 Carrier distributions far from equilibrium The injection of real e-h pairs into a semiconductor by means of short pulse optical excitation creates distributions which are, in general, far from equilibrium. If the band structure E ( k ) and the corresponding joint-density-of-states is known, the resulting initial distribution can be inferred directly from the light pulse spectrum, at least for small excitation. In the case of a direct-gap material with parameters E,= 1.5 eV, mr =O. 1 mo,mf = 0.5 mo, and with a 100 fs light pulse with a center energy of 2eV, the combineddensity-of-states is still relatively small, De-h 1020 cm-3 eV-’. With the known absorption coefficient of 5 x lo4 cm-’ this produces initial distributions of electrons with peak valuesf, 0.05 and holesf, 0.01 per Joule/cm2 excitation energy density. If the carriers were in thermal quasi-equilibrium, the saturation of interband transition because of the Pauli principle would occur at n,- lo” ~ m - This ~ . so-called “ bleaching” or “ Pauli blocking” of transitions has been observed in a series of elegant experiments (Chemla, 1985; Oudar and Miller, 1985; Knox et al., 1988a, 1988b).

-

Eo

*LE v k E(k+q)-E(k)-Ao+iAq

A direct consequence of Eq. (2-81) is that for longer distances the screened Coulomb potential of a point charge in an e-h plasma has characteristic oscillations in space with wavelength A- n/kF(Friedel, 1958). Treatments of screening which go beyond the framework of the so-called random-phaseapproximation (RPA) consider the distribution functionsf, ( k ) andf, ( k ) as dynamical variables. A self-consistency procedure leads to the concept of a dynamically screened dielectric function (Schafer, 1988; Haug, 1988; Axt and Stahl, 1994; Axt and Mukamel, 1998).

-

-

112


The relaxation behavior of optically excited non-equilibrium pair distributions in the intermediate density regime - ~ beenstudiedinpumplo’* ~ r n has probe experiments (Kash et al., 1989; Petersen and Lyon, 1990; Del Fatti et al, 1998). Coulomb scattering causes already at low densities of 10l6 cm-3 an extremely rapid spread of the width of the distribution in energy. The rate of increase in relative energy spread is of the order of 1 eV/ps for the parameters given above. The rates scale approximately with the initial excess energy A€ of the distribution (Bohne et al., 1990). For very short pulse excitation, within the Coulomb scattering regime, there is no more a qualitative distinction between semiconductors and metals: the initial dynamics of optically coupled pairs is similar in both cases. Under certain conditions, high-field transport may as well produce strong deviations from equilibrium. Selective population of subsidiary minima in the band structure (the Gunn-effect), and thresholds in electron-phonon coupling, or inter-band Auger processes may lead to peaked carrier distributions with very similar relaxation behavior. Qualitatively new phenomena are expected when the carrier distributions undergo rapid transients, either by application of a fast electric field pulse, or by optical injection with strong density gradients: the anisotropy in k-space leads to different time constants of energy and momentum relaxation, and causes the so-called velocity “overshoot”-effect (Seeger, 1999).

-

2.6.3 Carrier Dynamics on Ultrashort Time Scales Recent advances in the design of mode-locked solid-state lasers, notably the Ti : Sapphire system, have enormously

expanded the range of available pulse durations and energies. Stable ultrashort pulses with less than 10 fs duration have been demonstrated (Becker et al., 1988; Krausz et al., 1992) and optical spectroscopy with timeresolution in the 10 ... 100 fs range is now performed routinely. This breakthrough was the result of better control of dispersion effects in the laser cavity, and the improved understanding of self-phase-modulation (the optical Kerr effect) in the laser medium. It helped to optimize optical parametric oscillators to generate synchronized, fully coherent pulses with continuously tunable center wavelengths from 0.3 ... 3 p with comparably short widths. In the past decade, such ultrashort light pulses have been implemented into all branches of spectroscopy - like e.g. nonlinear transmission, reflectance and light scattering - and were applied likewise to semiconductors and metals (Schoenlein et a]., 1987, 1988; Lin et al., 1988; Del Fatti et al. 1998; Leitensdorfer et al., 1999; Shah, 1999). Time resolution down to a few femtoseconds corresponds to more than 100 Thz bandwidth in the frequency domain. Before the advent of ultrashort light pulses this range was not directly accessible. Wirebound electronic circuits reach their limit around 100 GHz, imposed by practical restrictions in geometrical size and the scaling of charge capacitance in small-scale devices (Kelly and Weisbuch, 1986). To get around this problem, electrooptic sampling techniques have been developed and have probed devices in-situ with very high spatial (- pn) and temporal (- 100 fs, mainly dispersion-limited) resolution. The longstanding frequency gap between coherent microwave and IR-spectroscopies has been successfully closed with photo-current driven Hertzian dipoles (Auston, 1988; Nuss et al., 1998), and also free-electron laser sources (Nordstrom et al., 1998). For


the first time the dynamics of carrier ensembles, like e.g. the Bloch oscillations in superlattice mini-bands, became directly accessible (Kelly, 1995). For the consistent description of the macroscopic electrodynamical response of a crystalline solid under intense optical excitation we have to go beyond the perturbative schemes of Sec. 2.2 and incorporate the quantum mechanical two-level behavior (i.e. saturation due to Pauli blocking) of the transitions. The latter concept had originally been developed for isolated two-level systems (spins, or atoms) interacting with an external driving B- or E-field (Bloembergen, 1962; Allen and Eberly, 1975). The modifications needed to describe a dense ensemble of atoms , Le. a solid, and its resonant interaction with light are contained in a now well established approach, cast in the form of the so-called semiconductor Bloch equations (SBE). For this purpose, the formidable many-body problem “ solid” can be treated only with drastic simplifications: one assumes isotropic eigen-values E ( k ) , and considers only two bands ( c , v) with “vertical” transitions connecting them (i.e. k, = k,), and discusses “optically coupled pairs” (see Fig. 2-21). The relevant observable quantities are polarization Pk= (&k+ uv,J for given k = k, = k,, the electron (resp. pair) occupation number n c , k = n k = (ac&’u c , J and the corresponding macroscopic quantities, namely dielectric polarization Ppair and concentration Npairof excited pairs:

113

Figure 2-21. Optical interband transitions in k-space seen in analogy with two-level atomic systems (see text).

/’

IE”

\

analogy to the Bloch equations describing a two-level isolated atomic system (Haug and Koch, 1994): Pk=ek Pk-i

(1-2

nk)

Dk

(2-84a) (2-84b)

Here the external control parameter is the driving field E, represented by the Rabi frequency Qo=d,, E, where d,, defines the interband coupling strength. The semiconductor is described by its k-dependent transition energies &k, and the matrix element Vq l / q 2 accounts for Coulomb scattering between (pairs of) excited carriers. The two quantities E and &k are renormalized through the excitation (resp. carrier density) dependent Coulomb scattering. In mean field approximation, i.e. neglecting spatial correlations among the carriers, one obtains

-

&k = Egap

h2 k 2 +-

$2k =

+

2m*

vq Pq-k

c

vq nq-k

(2-85) (2-86)

9

(2-82) (2-83) The electric field E is treated here classically. In the simplest case, the SBE are derived within the framework of Hartree-Fock base states (see Sec. 2.2 above), and in close

Setting all terms with n k , Pk n k , and p t in Eq. (2-84)-(2-86) to zero, Le. in the limit of vanishing excitation, the SBE reduce to the Schroedinger equation for one e-h pair, and reproduce the exciton states discussed in Sec. 2.2.3 above. The SBE describe, in the spirit of the mean field approximation, the full dynamics of optical excitations for arbitrary

114

Int.


’ e

A /

\

d o

-_---incoherent

\

k

0

,!‘:I

coherent emission

I/ I

0

,J

\

t

0

Figure 2-22. Dynamics of interband polanzation, pair density and coherent emission intensity in the framework of the semiconductor Bloch equations (schematic. see text)

degrees of optical excitation in semiconductors. The time response of the local polarization Pk (r) is not instantaneous, but has a memory of the form t

Pk a

j

dr’ E (f’)

(2-87)

--m

and gives rise to a wealth of coherent phenomena. In experiments, the mutual coherence between the driving light field and the optically coupled electron-hole pairs was demonstrated for the first time in the context of the optical (AC) Stark effect with A w < E , (Froehlich et al., 1985; Mysyrowicz et a]., 1986), and for polarization wave

scattering of a single exciton resonance (Wegener et al., 1990). Coherent pulse propagation and beating phenomena between different exciton resonances, as well as the optical response in the continuum, tz o>E,, were explored in theory (Stahl and Balslev, 1986; Axt and Stahl, 1994). Since then many transient optical experiments at isolated excitonic resonances have been consistently described within the framework of the semiclassical SBE : photon echoes (No11 et al., 1990), Rabi oscillations (Cundiff et al., 1994; Schulzgen et al., 1999), self-induced transparency (Giessen et al., 1998), and the momentum spread in tightly focussed excitation spots (Hanewinkel et al., 1999). Interference effects, level shifts and polarization selection rules were investigated in detail in theory and experiment (Joffre et a]., 1989; Knox et al., 1989; Henneberger and Schmitt-Rink, 1993; Sieh et al., 1999). At the level of Eq. (2-84), the calculated optical spectra give sharp &like resonances, because they connect unbroadened k,and k,-eigenstates which provide no dephasing mechanism. To account for the empirically observed dephasing and damping phenomena (see Fig. 2-22), purely phenomenological damping terms with relaxation times T , and T 2 , in analogy with spin systems, were usually added ad hoc to Eqs. (2-84a, b). One tried to include in a qualitative way the effects of dephasing mediated by Coulomb scattering. For incident frequencies above-gap, fi u >E,, qualitatively different effects are expected. A coherent light pulse will create an initial electron-hole pair polarization P,, (see Sec. 2.2) with well-defined temporal and spatial coherence. But during and after build-up of P,, the phase correlation between the driving field and the induced polarization is rapidly relaxed by momentum scattering due to Vq: in the continuum

2.8 References

of e-h pair states, the Coulomb interaction does not “filter” out the discrete spectrum of bound pair states. It does not keep any long-term phase coherence, but dephases all pair excitations on a very short time scale. The transition optically driven and coherently evolving pair states into the regime of random motion of carriers in a bipolar e-h plasma is of great practical interest in the context of quantum confined optoelectronic devices and their coupling to the environment. Their information processing potential depends crucially on the interplay between both regimes. The former is closely connected with ballistic transport, the latter is that of conventional charge transport, with no long-term phase correlation between carriers. The field of nonlinear optics in semiconductors is rapidly developing. It is evident that inhomogeneous broadening due to static disorder and low-lying vibronic excitations will affect qualitatively even the linear optical response on the 1 ... 100 ps-scale (Langbein et al., 1999). Nonlinear optical spectroscopies of exciton ensembles in real, i.e. disordered semiconductor structures, and their interpretation beyond the simple homogeneous SBE model discussed here, begin to emerge (Euteneuer et al., 1999). Many details of experimental observations are now interpreted - and sometimes reinterpreted - beyond the purely phenomenological longitudinal and transverse relaxation time concept. One tries to go beyond the SBE description, in order to include exciton correlations and their intrinsic dephasing mechanism (Kner et al., 1998; Ostreich and Sham, 1999), and to cover quantum kinetic effects in the Coulomb scattering process (Bonitz, 1998; Wehner et al., 1998; Huge1 et al., 1999).

115

2.7. Conclusion It should be noted that the concepts which have been discussed in this article were all developed in the context of ideal, homogeneous, bulk crystalline solids. Their immediate application to semiconductors, and especially to nanostructures with mesoscopic length scales, necessarily requires caution and modifications. Some of the assumptions were made for the sake of simplicity and cannot represent realistic experimental conditions. At present, the views on nonlinear charge transport in small semiconductor structures and their optical properties on the ultrashort time scale begin to converge. The concepts of ballistic transport of electrons on one hand and coherent electron-hole pair dynamics on the other are closely related to each other, but are usually presented in very different formalisms. A lot has still to be done to fill this gap. It is safe to say that the field of optics and charge transport in small scale structures is far from being mature.

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Wigner, E. P. (l932), Phys. Rev. 40, 749. Wilson, W. K., Glytsis, E. N., and Gaylord, T. K. (1993), IEEE Journal QE 29, 1364. Wingreen, N. S., Stanton, C. J., and Wilkins, J. W. (1986), Phys. Rev. Lett. 57, 1084. Woggon, U. (1997), Optical Properties of Semiconductor Quantum Dots, Berlin: Springer. Wooten, F. (l972), Optical Properties ofsolids. New York: Academic Press; Ch. 5. Yablonovitch, E., Allara, D. L., Chang, C. C., Gmitter, T., and Bright, T. B. (1986), Phys. Ret: Lett. 57, 249. Yacoby, A , , Heiblum, M., Mahalu, D., Shtrikman, H. (1995), Phys. Rev. Lett. 74, 4047. Yoffa, Ellen J. (1980), Phys. Rev. B21, 2415. Yu,P. Y., and Cardona, M. (1996), Fundamentals of Semiconductors. Berlin: Springer. Zakharchenya, B. P., and Meier, F. (1984), Spin Orientation. Amsterdam: North Holland. Ziman, J. (1963), Electrons and Phonons. Oxford: Oxford University Press; Ch. 7. Zimmermann, R., and Runge, E. (1997), Phys. Stat. Sol ( a ) 164, 5 1 1. Zimmermann, R. (1990), in: Advances in Solid State Physics Vol. 30. Braunschweig: Vieweg. Zimmermann, R., Kilimann, K., Kraeft, W. D., Kremp, D., and Roepke, G. (1978), Phys. Star. Sol ( b )90, 175. Zschauer, K. H. (1973) in: Gallium Arsenide and Related Compounds, Bristol: Institute of Physics. p. 3.

3 Intrinsic Point Defects in Semiconductors 1999

.

George D Watkins Department of Physics. Lehigh University. Bethlehem. U.S.A.

123 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2 3.2.1 Defect Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 127 3.2.2 The Silicon Vacancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2.2.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Effects of Lattice Relaxation - General Considerations . . . . . . . . . . . 129 3.2.2.3 Electrical Level Positions of the Vacancy . . . . . . . . . . . . . . . . . . 130 132 3.2.2.4 Vacancy Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.5 Vacancy Interactions with Other Defects . . . . . . . . . . . . . . . . . . 133 3.2.3 The Silicon Interstitial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 135 3.2.3.1 Interstitial Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2 Trapped Interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 138 3.2.3.3 Theory of the Silicon Interstitial . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Close Frenkel Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Other Group IV Semiconductors . . . . . . . . . . . . . . . . . . . . . 141 3.3 3.3.1 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.3.2 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.3.3 Sic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11-VI Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 ZnSe 143 3.4.1.1 The Zinc Vacancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.4.1.2 Zinc Vacancy Diffusion and Interaction with Other Defects . . . . . . . . . 145 3.4.1.3 Interstitial Zinc and Close Frenkel Pairs . . . . . . . . . . . . . . . . . . . 145 146 3.4.1.4 Defects on the Selenium Sublattice . . . . . . . . . . . . . . . . . . . . . 147 3.4.2 Other 11-VI Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2.1 The Metal Vacancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.4.2.2 The Chalcogen Vacancy . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.4.2.3 The Other Intrinsic Defects . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.4.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.5 111-V Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.5.1 Antisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 149 3.5.1.1 The Anion Antisite VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.2 The Cation Antisite 111, . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.5.2 Group-I11 Atom Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.5.3 Metal Interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

122

3.5.4 3.6 3.6.1 3.6.1.1 3.6.1.2 3.6.1.3 3.6.2 3.6.2.1 3.6.2.2 3.6.2.3 3.6.2.4 3.7 3.8


Defects on the Group-V Sublattice . . . . . . . . . . . . . . . . . . . . . . Summary and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Group IV Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 11-VI Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11- V Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Migration Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 154 155 155 155 156 156 157 159 160 160 160 161


List of Symbols and Abbreviations a1

A1 C3" D2d

e e

eD

E E, EJT E, E( i l j ) g

h+ 1

I kB

kQ S

S T t2

T2 Td

U ua

V VQ V W Y

n

e CJ 0,

z

cc DLTS ENDOR EPR

symmetric representation for a molecular orbital symmetric representation for a many-electron state trigonal symmetry tetragonal symmetry doubly degenerate representation for a molecular orbital electron electron bound to a shallow donor doubly degenerate representation for a many-electron state electron energy at the conduction band edge Jahn-Teller relaxation energy electron energy at the valence band edge Fermi level position at which defect changes from charge state i to j gyromagnetic ratio for a paramagnetic defect hole interstitial (subscript) interstitial Boltzmann constant spring force constant for distortion mode -Q substitutional (subscript) electronic spin absolute temperature (K) triply degenerate representation for a molecular orbital triply degenerate representation for a many-electron state tetrahedral symmetry Hubbard correlation energy activation energy (barrier) for defect migration group-V atom linear Jahn-Teller coupling coefficient to distortion mode Q vacancy defect formation energy frequency of a rate limited process polarization parallel to a defect symmetry axis resistivity polarization perpendicular to a defect symmetry axis capture cross section for electrons capture cross section for holes time constant for a rate limited process configurational coordinate deep level transient capacitance spectroscopy electron-nuclear double resonance electron paramagnetic resonance

123

124

LEC LVM


liquid encapsulated localized vibrational mode MBE molecular beam epitaxy ODENDOR optical detection of electron-nuclear double resonance ODEPR optical detection of electron paramagnetic resonance OMVPE organometallic vapor phase epitaxy

3.1 Introduction

3.1 Introduction

125

growth and processing of materials, but they must result from non-equilibrium proThis chapter is an updated, and substancesses such as aggregation and precipitation tially revised edition of a review chapter of the quenched-in intrinsic defects, etc. originally presented in Volume 4 of the The fact that vacancies and interstitials Materials Science and Technology series are thermodynamically stable has impor(Watkins, 1991). It attempts to correct some tant consequences in high temperature proof the misinterpretations and uncertainties cessing of materials in that they supply the at that time, and bring up to date (1/1/99) principal mechanisms for mass transport what is now currently believed to be known and diffusion. With an activation barrier concerning the intrinsic point defects in U,for the migration of the defect, its consemiconductors. tribution to diffusion is proportional to In this chapter, we will define “intrinsic exp [- ( W + U,)lk,TJ. The migration barrier U, and the formation energy Ware therefore point defects” to mean the simplest structuimportant properties that we would like to ral defects that can be formed in a crystalline lattice by rearrangements of the host establish for each defect. atoms, no impurities being involved. With In a semiconductor, the intrinsic defects are of interest for a number of other importhis definition, there are three basic types: (1) a lattice vacancy (a missing host atom); tant reasons as well. Because a defect per(2) an interstitial (an extra host atom turbs the periodicity of the perfect lattice, squeezed into the lattice); and (3) an antisits presence can introduce electronic states ite (atom A on site B, or vice versa, in an in the forbidden gap of the material. We AB compound crystal lattice). These in turn therefore need to understand its electronic can, of course, be considered the elemenstructure as well - its available charge states, the local lattice relaxations around it, tary “building blocks” for all other more complex structural defects - divacancies, and how these affect its electrical and optical properties, its stability and mobility, and trivacancies, voids; interstitial pairs, interstitial aggregates, etc. Even dislocations, its interactions with other defects. grain boundaries and external surfaces can And so the study of intrinsic defects in be considered to be constructed from arrays semiconductors is a very rich, complex, and often frustrating enterprise. Fragments of of these three fundamental units. Vacancies, interstitials, and antisites are information and clues have been assembled in all of the elemental and compound also in a sense the only true intrinsic defects in that a significant concentration of each semiconductors after many years of study. However, in many cases they are only clues is thermodynamically stable at any temand the interpretations are often controverperature. [With its formation energy W j , sial. Even when a consensus seems to have the fractional concentration of defect j is given approximately by exp (-Wj/kB7‘)l.occurred, our previous experiences in such matters urge caution, for it could well be The binding energy between defects to wrong. The principal problem is identificaform complexes is invariably significantly tion of the defect being studied. smaller than the individual defect formation In the case of vacancies and interstitials, energies so that the thermodynamic equilibtherefore, the first question that has had to rium concentration of complexes becomes be addressed experimentally is how to provanishingly small as the size grows. Comduce them and know that it is their properplexes can and do form, of course, in the

126


ties that are being studied. In semiconductors, attempts to quench them in by rapid cooling from elevated temperatures, often successful in metals, has invariably failed due to the combination of high diffusional mobility of the defect and its low concentration near the melt, plus the ubiquitous presence of fast diffusing trace impurities from the ambient, or already present in the material, that tend to dominate the material property changes. The only clean, direct, and unambiguous way that has been found to produce the defects is by high energy (> 1 MeV) electron irradiation at cryogenic temperatures (Corbett, 1966). In this way, the simple interstitial and vacancy produced by a Rutherford “collision” of electron with a host atom nucleus can be frozen into the lattice for study. The scenario of such an event is shown in Fig. 3-1. As the temperature is subsequently raised, one of the species will become mobile first. At this stage, some of the pairs will recombine, while for others, the mobile species will escape and diffuse through the lattice until trapped by other defects to form new pair defects. At a higher temperature stage, the remaining partner becomes mobile and is trapped also by other defects.

._---.

-e-‘\ .x Figure 3-1. An electron irradiation damage event. Recovery occurs by a) vacancy-interstitial recombination, b) interstitial migration until trapped, c) vacancy migration until trapped.

Secondly, identification requires experimental observation techniques that reveal the structure of the defect at the atomic level. The only techniques that fully satisfy this requirement are the magnetic resonance ones. These include electron paramagnetic resonance (EPR) (Watkins, 1975 a, 1976a, 1998), electron-nuclear double resonance (ENDOR) (Ammerlaan et al., 1985), and optical detection of EPR (ODEPR) (Cavenett, 1981) and ENDOR (ODENDOR) (Spaeth et al, 1992). The unique information provided by these techniques, when they work, is anisotropy of the Zeeman (9) and fine-structure ( D ) interactions, which reveal the defect symmetry, and resolved hyperfine interactions with the atomic nuclei in the core and neighborhood of the defect, which allow a detailed mapping of the atoms involved and their structural arrangements. Recently, local vibrational mode (LVM) spectroscopy has also been demonstrated in a few cases to provide reliable defect information for intrinsic defects trapped by light impurity atoms. In these cases, isotope shifts, and symmetry determination by shifts and alignments under applied uniaxial stress, supply the identifying features. Once the defect is identified, then electrical, optical, and other measurements can be performed in close correlation with the magnetic resonance experiments, to further establish its properties. In this chapter we therefore plan to deal only with those intrinsic defects for which magnetic resonance and LVM studies have supplied relatively firm identifications. The best documented system in this regard is silicon. Here, the isolated vacancy has been observed, the electrical and diffusional properties of several of its charge states determined, and a host of its interactions with impurities and other defects identified. The isolated interstitial has not been directly observed. However, a microscopic iden-

3.2 Silicon

tification of its interactions with other defects has served to reveal many of its properties. In this case, theory has served also to fill in some of the important gaps. Information is also beginning to trickle in for the other important group IV semiconductors diamond and the common polytypes of Sic. The next best understood materials are the 11-VI semiconductors. In ZnSe, for instance, isolated interstitials and vacancies and close Frenkel pairs on the metal sublattice have been identified and studied in substantial detail. In several of the other 11-VI compounds, metal vacancies with similar characteristics have been detected and clues to the properties of the chalcogen vacancy have also been obtained. Finally, there are the 111-V compounds. Here the important role of the antisites has clearly been established by magnetic resonance studies although there remain many controversies about their properties and their interactions with other defects. There are fewer hard facts about vacancies and interstitials, however. The isolated gallium vacancy in GaP appears to have been identified and magnetic resonance of a few defects believed to be associated with gallium interstitials have been reported in Gap, AlGaAs, and GaN. Otherwise, there are no firm identifications. In what follows, therefore, we will restrict ourselves to what has been learned in these three systems, with major emphasis on silicon and ZnSe, the best documented within the three. What we have learned from the elemental semiconductor silicon and the substantially ionic 11-VI compound semiconductors can hopefully serve as a guide to interpreting results in the more covalent 111-V compounds which they bracket and for which magnetic resonance methods have so far been less successful.

127

3.2 Silicon 3.2.1 Defect Production A most surprising and unexpected result is found when p-type silicon is irradiated with 1.5-3.0 MeV electrons at 4.2 K. EPR spectra of isolated vacancies are observed but none that can be identified with interstitials or close Frenkel pairs. Instead spectra of defects identified with interstitials trapped by impurities are observed with approximately 1 : 1 intensity to those of the isolated vacancies. The unavoidable conclusion is that silicon interstitials must be mobile, migrating long distances even at 4.2 K! (Watkins, 1964). In terms of the expected scenario of Fig. 3-1, process (b) is occurring even during the 4.2 K irradiation and only isolated vacancies and trapped interstitials are observed. We will return to this surprising result later when we discuss the interstitial in more detail. Let us at this point accept the fact and concentrate on what has been learned about the isolated vacancy.

3.2.2 The Silicon Vacancy 3.2.2.1 Electronic Structure The vacancy is observed to take on five V-, charge states in the forbidden gap (V, Vo, V+,V++).Two of these (V' and V - ) , having an odd number of electrons, are paramagnetic and have been identified directly by EPR (Watkins, 1964). Using these two spectra as probes, it has been possible to develop a simple model which accounts in remarkable detail for the electronic structure of all the charge states (Watkins, 1975b, 1986). This is illustrated in Fig. 3-2. Here the electrons are depicted as occupying highly localized molecular orbital states made up primarily as linear combinations of the broken bond orbitals (a, b, c, d) of the

128

a)


Figure 3-2. Simple one-electron molecular orbital model for the vacancy in silicon (Watkins, 1983).

vtt atbtctd Td

b) V*

(12 I

(a,)

Tib:; -

r*(o)l

Td

c) vo

(e

I

a - b t c -d ( a + b c - d) a - b- c t d

-

atbtctd

D2d

-(el a-b-ctd atbtctd

c-b a-d (0;I (0;

D2d

I

-atd -btc

c2V

four silicon atoms surrounding the vacant site. An undistorted vacancy has full tetrahedral Td symmetry and simple group theory arguments require that a singlet (a,) and triplet (t2) set are formed. The fully bonding a , state will be the lowest in energy, as shown, due to the attractive Coulomb interaction of the electrons with the positive cores of the four silicon neighbors (the “crystal-field”). For V + + , Fig. 3-2a, two electrons occupy the a, state, spins paired. The defect is diamagnetic and no EPR is observed. For V’, Fig. 3-2 b, the third electron goes into the t2 orbital. Because of the degenera-

cy associated with the orbital, a tetragonal Jahn-Teller distortion occurs (Jahn and Teller, 1937; Sturge, 1967), lowering the total energy of the defect with atoms a and d, and atoms b and c, pulling together by pairs, as shown. The resulting orbital (b2 in reduced D2, symmetry) contains the unpaired electron which is spread equally over the four silicon atoms, as has been established in the EPR studies from resolved 29Si hyperfine interactions at each site. The existence of the tetragonal distortion is evidenced by small tilts in the dangling bond directions which are detected in the hyperfine interactions, as well as by anisotropy in the g-values of the center.

3.2 Silicon

In forming V o , Fig. 3-2c, the next electron goes also into the b, orbital, spin paired, further enhancing the tetragonal Jahn-Teller distortion because now the energy gain of two electrons is involved. In this state the defect is diamagnetic and no EPR is observed. For V-, Fig. 3-2d, the fifth electron goes into the degenerate e orbital and an additional Jahn-Teller distortion occurs. This distortion is of b, symmetry (in D2d) with atoms b and c further pulling together and a and d separating slightly. As a result, the unpaired electron ends up on atoms a and d, again as established by EPR. For V=, the absence of EPR suggests again a diamagnetic state, the sixth electron perhaps going paired off into the a-d orbital. (This is not shown in the figure, however, because so far no direct or indirect evidence of its symmetry has been obtained. It cannot be ruled out that a completely different set of relaxations set in for this charge state.) For the two paramagnetic charges states (V’ and V-), analysis of the 29Si hyperfine interactions indicates that 60-65% of the wavefunction is accounted for in these dangling orbitals. In the case of V-, ENDOR studies have resolved weaker 29Sihyperfine interactions with an additional 26 inequivalent shells of neighboring silicon atom sites accounting for most of the remaining wavefunction (Sprenger et al., 1983). The true wavefunctions therefore of the states derived from the t, orbitals can be considered to have -60-65 % of their density highly localized in these dangling orbitals with a weak tail, extended over many shells of neighbors, accounting for the remaining 35 -40%. The remarkable success of these simple one-electron models reveals that the electron-electron interactions that tend to favor parallel spin coupling (Hund’s rules for

-

-

129

atoms, etc.) are small. We are, in effect, in the strong crystal-field regime. We fill each level before going to the next. When degeneracy occurs, a Jahn-Teller distortion occurs which imposes a new crystal field, decoupling the electrons again. It is by no means obvious that this necessarily should be the case. There has been much controversy in the theoretical literature through the years on this point, with no clear consensus (Stoneham, 1975; Surratt and Goddard, 1978; Lannoo and Bourgoin, 1981; Malvido and Whitten, 1982). And so it is really only through experiment that we know that this simple case applies in silicon, as later justified by theory (Lannoo et al., 1981; Lannoo, 1983). We must be prepared for other possibilities in other systems, as we shall see in some of the larger bandgap elemental and compound semiconductors.

3.2.2.2 Effects of Lattice Relaxation General Considerations When lattice relaxations such as these Jahn-Teller distortions occur at a defect, significant consequences may result in terms of the defect’s electrical and optical properties, and its stability. Since this is a feature applicable to many of the intrinsic defects that will be discussed in this chapter, not just the silicon vacancy, let us now develop some of the general concepts. In Fig. 3-3, we show a “configurational coordinate’’ (CC) diagram for a hypothetical defect D which undergoes a distortion (mode Q)in capturing an electron to become neutral, Do. Shown are three total energy surfaces, one for the initial D’ state, a second displaced by the band gap Eg to represent D’ plus a free electron (e-) and hole (h’), and a third when the defect traps an electron, Do + h+,and distorts. As illustrated in the figure, the electrical level position measured with respect to the

130


\

/

-0

Figure 3-3. Configurate coordinate (CC) model for a defect with large lattice relaxation in a semiconductor (Watkins, 1983).

conduction band edge is the energy difference between the neutral charge state Do + h’ and the ionized state D+ + e- + h+ each determined in its ful1.v relaxed state. Jahn-Teller distortions therefore directly alter the electrical level positions in the gap. Secondly, they cause Stokes shifts for optical ionizing transitions which are “vertical” on the diagram, causing differences between ionization energies determined optically and electrically. (The true electrical level position corresponds to the “zero phonon line” of the optical transition.) Third, the intersecting energy surfaces supply a “multi-phonon’’ mechanism for electron (a,)and hole (ah)capture processes, which are otherwise difficult for a deep level. When the distortion is large, as depicted in the figure, they provide in addition a radiationless recombination path for electrons and holes, the neutral state providing a “short circuit” across the gap by alternate electron and hole capture. Finally, carrier capture means entry into a high vibrational state of the new charged state and this burst of energy may serve to assist the defect over its diffusional barrier, a phenomenon called recombination-enhanced migration (Kimerling, 1978: Bourgoin and Corbett, 1978).

3.2.2.3 Electrical Level Positions of the Vacancy Fig. 3-4 summarizes what is presently known about the level positions for the vacancy (Watkins, 1986). The levels in Fig. 3-4a represent a consensus of theoretical estimates for the single particle t, and a, levels for the undistorted neutral vacancy (Baraff and Schluter, 1979; Bernholc et al., 1980). These can be considered to correspond to the t2 and a, orbitals of Fig. 3-2. In Fig. 3-4b we have included corrections to the single particle neutral t, gap level positions vs. charge state to account for the Coulomb repulsion energy between the electrons as the occupancy of the t, level increases. The level separation U (the Hubbard correlation energy), taken in the figure to be -0.3 eV, is a typical value found experimentally for deep levels in silicon and agrees well with the theoretical estimates for the vacancy (Baraff et al., 1980a).

(a)

@)

(c)

Figure 3-4. Electrical level positions of the vacancy in silicon: a) Calculated single particle levels for unrelaxed p. b) Corresponding estimates for the level positions. c) Experimental results, after Jahn-Teller relaxations (Watkins, 1983). (Level positions in eV from the valence band edge, ,Ev).

3.2 Silicon

In Fig. 3-4c, we show the results after the Jahn-Teller distortions. Each level drops in the gap because with each added electron the Jahn-Teller distortions increase, Fig. 3-2. The first E(O/+) and second E (+/++) donor level positions have been estimated directly in EPR and correlated deep level transient capacitance spectroscopy (DLTS) studies (Watkins and Troxell, 1980; Newton et al., 1983; Watkins, 1986). The levels have also been measured and confirmed subsequently via Hall measurements (Emtsev et al., 1987). The results are indicated with respect to the valence band edge in Fig. 3 - 4 where ~ the Hall measurement estimate for the first donor level position (0.03 eV) has been selected, rather than the hole emission kinetics result from the EPR studies (0.05 eV). We note here a remarkable result: The levels are reversed, in negative- U ordering (Watkins, 1984), the first donor state E(O/+) being deeper than the second E(+/++).The gain in Jahn-Teller energy in going from V +to Vo is apparently great enough to overcome the Coulomb repulsion energy between the electrons so that the ionization energy for VO + V+ + e- is 0.10 eV greater than that for the removal of the second electron V+ + V+++ e-. (Alternatively, the 0.13 eV ionization energy to remove the first hole V + + V + h + is greater than the 0.03 eV required for the second hole V + + Vo+h'.) There is in effect a net attraction between the two electrons (holes) at the defect. This effect was first predicted theoretically for the vacancy by Baraff et al. (1979, 1980a, b) and subsequently confirmed in detail by experiment. It can be understood in a very simple way. We outline this below because, with it and the experimental level positions, a direct estimate of the magnitude of the Jahn-Teller energies is possible (Baraff et al., 1980b).

131

For n = 0, 1, or 2 electrons in the b2 orbital of Fig. 3-2 b and c, the relaxation energy of the corresponding charge state ( V++, V+, or Vo, respectively) can be approximated as

E (It) = - It VQ + (1/2) kQ Q2

(3-1)

where V, is the linear Jahn-Teller coupling coefficient for the b2 orbital to the tetragonal distortion mode, Q is it amplitude, and k, is the force constant describing the elastic restoring forces on the atoms for this mode. (-V, Q is therefore the lowering of the b2 orbital, which in this simple independent electron model is independent of occupancy). Minimizing Eq. (3-1) with respect to Q gives for the Jahn-Teller stabilization energy

EJT( n ) = - n 2 V2Q / kQ ~

(3-2)

The E (+/++) level is therefore lowered from its relaxed state by E j T ( 0 ) - EJT(1) = Vi/2 k , and the E (O/+) level is lowered by EJT(1) - EJT(2) = 3 Vg/2 k,. From this, U,, = E (()/+)-E (+/++) =

u- Vi/kQ

(3-3)

which is negative if Vi/k, > U , the criterion originally derived by Anderson (1 975). Setting U,, to the experimental value of -0.10 eV and with U = 0.3 eV, Eq. (3-3) gives Vilk, = 0.40 eV. With Eq. (3-2), the Jahn-Teller relaxation energies for V+ and Vo become 0.2 eV and 0.8 eV, respectively. The Jahn-Teller distortions of Fig. 3-2 are therefore not just interesting curiosities that describes subtle features of the electronic structure. They are large, being comparable to the band gap with serious consequences for the electrical level positions as well. We will see that they also have important bearing on the diffusional mobility of the vacancy. No information is presently available on the positions of the acceptor levels E(-IO) and E(=/-). The failure to detect DLTS lev-

132


els associated with them in n-type material has been interpreted as indicating that they must be greater than -0.17 eV below the conduction band edge (Troxell, 1979).

3.2.2.4 Vacancy Migration As the temperature is raised, the vacancy EPR spectra disappear and a large assortment of new spectra emerge which have been identified as vacancies paired off with impurities. The annealing therefore is the result of long range migration of the vacancy. Detailed study of the kinetics via EPR and DLTS have given the following results: In low resistivity n-type silicon, where the vacancy is in the V = charge state, the activation energy has been determined to be 0.18 0.02 eV (Watkins, 1975b). In low resistivity p-type material with the vacancy in the Vc+ state, the energy is 0.32 k 0.02 eV (Watkins et al., 1979). Under reverse bias in DLTS studies, the activation energy for the annealing process was found to be 0.45 k 0.04 eV, presumably the property of V o (Watkins et al., 1979). These results are summarized in Table 3-1. Such low activation energies for vacancy migration are indeed surprising for a material which doesn’t melt until -1450°C and for which activation energies of selfTable 3-1. Activation energies for the silicon vacancy migration.

Conductivity type

Charge state

U,eV)

Loa e n-type High Q p-type (re\erse bias) Loa Q P-tYPe

V=

0.18 f 0.02

VO

0.45 5 0.04

V++

0.32 & 0.03

diffusion are -4-5 eV (see Chap. 5 of this Volume). The unavoidable conclusion is that for the vacancy contribution to diffusion, the major part of the activation energy W + U , must be coming from the formation energy W. Theoretical ab initio local density calculations have confirmed that vacancy formation energies are indeed large, the most recent estimates ranging from 3.3 eV (Pushka et al., 1998) to 4.1 eV (Blochl et al., 1993). Also, high temperature quantum molecular dynamics calculations of the same sophistication match the high mobility and low -0.4 eV barriers (Smargiassi and Car, 1990; Blochl et al., 1993). Therefore, this result, which has been difficult to accept over the many years since it was first experimentally established, is most certainly correct (see Chap. 5 of this Volume). It has also been established that vacancies can be made to migrate through the lattice at temperatures well below those required for thermally activated diffusion, under conditions of minority carrier injection or optical excitation. Detailed studies have been performed in both n- and p-type silicon which demonstrate unambiguously athermal (not thermally activated) migration at 4.2 K under such conditions and have served to estimate the efficiency of the process (Watkins et al., 1983; Watkins, 1986). As pointed out in Section 3.2.2.2, and in Fig. 3-3, Jahn-Teller distortions provide a natural mechanism for such recombinationenhanced processes where the energy associated with carrier capture or “vertical” optical excitation is funneled into local lattice vibrations which assist the defect over its migration barrier. In the case of the vacancy, where the estimated Jahn-Teller energies exceed the small diffusional barriers, no additional thermal energy is required and the process is athermal.

133

3.2 Silicon

3.2.2.5 Vacancy Interactions with Other Defects Shown in Fig. 3-5, in addition to the isolated vacancy, are the structures deduced for the various identified trapped vacancies, as well as the electrical level positions that have been determined for them. When trapped by substitutional tin, the large tin atom moves into a position halfway into the vacancy, producing for the neutral state what can be considered as a tin atom in the center of a divacancy, as illustrated in Fig. 3-5c (Watkins, 1 9 7 5 ~ ) Levels . at E, + 0.32 eV and Ev + 0.07 eV have been identified as its first E (O/+) and second E (+/++) donor levels, respectively (Watkins and Troxell, 1980). Substitutional oxy-

-

+ 0.05-

7-7

9 . ) (V . Ge)'

/ / / /

h.) (V . Ge)'

gen (Fig. 3-5 d) results from the trapping of a vacancy by interstitial oxygen, a common impurity in silicon. On-center substitutional oxygen would normally be expected to be a double donor. However, EPR and LVM experiments show that it moves off-center, bonds to two of the silicon atoms and the defect becomes instead a single acceptor with its E ( 4 0 ) level at E, - 0.17 eV (Watkins and Corbett, 1961; Corbett et al., 1961; Bemski, 1959; Brower, 1971, 1972). Vacancies trapped by the substitutional group-V atoms P (Watkins and Corbett, 1964), As, or Sb (Elkin and Watkins, 1968) take on the configuration shown in Fig. 3-5 e, introducing a single acceptor level at Ec - 0.43, E, - 0.47, or E, - 0.44 eV, respectively (Troxell, 1979). The divacan-

+ 0

-0.07 /+?I

0

+

0.32

'/

0.2 1

/ / / / / / / / / / / / 1 EV

y@ 98 d%

i.) (V AI)-*

j.) ( v .

u)'

k.) (V

H2)'

d-b I.) (V ' H)"

Figure 3-5. Structures of isolated and trapped vacancies in silicon, and their identified electrical levels (eV from the nearest band edge). Jahn-Teller distortions are evidenced by the bond reconstruction by pairs. The localization of the unpaired electron observed by EPR is indicated in black (Watkins, 1999).

134


cy results by pairing of two vacancies during vacancy anneal and its structure is shown in Fig. 3-5f (Watkins and Corbett, 1965; deWit et al., 1976; Sieverts et al., 1978). [It has also been observed to form directly as a primary event in the irradiation with a somewhat higher threshold energy than that for single vacancy formation (Corbett and Watkins, 1965)]. It takes on four charge states (VV+, VVo, VV-, VV') and has been observed by EPR in the single plus and minus charge states. Its second acceptor level E(=/-) has been determined to be at E, - 0.23 eV, its first acceptor level E ( 4 0 ) at E, - 0.41 eV, and its single donor state E (O/+) at E, + 0.21 eV (Kimerling et al., 1979). In the case of the vacancy trapped by substitutional Ge, its dangling bond reconstructions, shown in Fig. 3-5g and h, reveal an only slightly perturbed vacancy (Watkins, 1969), and its general behavior vs Fermi level, optical excitation, etc., appears identical to that of the isolated vacancy. This suggests a very similar level structure (including negative-U), as indicated in the figure for the isolated vacancy. A level presumed to be its E (=/-) double acceptor level has been reported at E, - 0.29 eV (BudtzJorgensen et al., 1998). The aluminum-vacancy pair has been studied only i n a photo-excited state which has the configuration shown in Fig. 3-5i (Watkins. 1967). A level at E , + 0.52 eV has been tentatively associated with it (Troxell, 1979). The boron-vacancy pair differs from the others in that the boron atom appears to prefer the next-nearestneighbor position to the vacancy (Fig. 3-5j) (Watkins. 1976b; Sprenger et al., 1985). The level positions for it have not been established. Shown also in Fig. 3-51 and k are the structures deduced for a vacancy containing one (Bech Nielsen et al., 1997) and two

(Bech Nielsen et al., 1995; Bai et al., 1988; Chen et al., 1990) hydrogen atoms, respectively, which have also been identified by EPR techniques. [The EPR identification of V-H, (Chen et al., 1990) has been challenged (Stallinga and Bech Nielsen, 1998), and is a subject of current controversy (Chen et al., 1998).] Combined with the rich variety of hydrogen LVM absorption spectra observed in hydrogen-implanted material, it has been possible to detect and identify the signatures of the complete set, V . H, V-H,, V.H,,and V.H,(BechNielsenetal., 1995). The electrical levels have not been determined for any of these, but, as suggested by their structures, the electrical property of V . H, can be reasonably expected to be similar to that for V . 0 (Chen et al., 1990), and similarly that of V - H to that of V.P (Bech Nielsen et al., 1997). For most of these defects again simple one-electron molecular orbital models similar to that of Fig. 3-2 for the vacancy (but with reduced symmetry due to the nearby defect) provide a satisfactory description of their structures. For the ground states, the levels are populated according to the large crystal field regime and, when degeneracy remains, Jahn-Teller distortions occur, decoupling the electrons leading to minimum multiplicity (S = 112 or 0) as for the isolated vacancy. The only exception is for the neutral tin-vacancy pair, which has an S = 1 ground state. In Fig. 3-5, Jahn-Teller distortions are evidenced by the bond reconstruction by pairs. The localization of the unpaired electron observed by EPR is indicated in black. As for the isolated vacancy, hyperfine interactions again indicate 65 % localization on the atoms adjacent to the vacancy. In the case of the divacancy (Cheng et al., 1966) and the group-V atom-vacancy pairs (Watkins, 1989), related optical absorption bands have also been reported. These have

-

3.2 Silicon

I

I

135

1

,

V.H

r’l

V.HI

been interpreted in terms of transitions between occupied and unoccupied molecular orbital states for the centers, providing further confirmation for the molecular orbital models. In addition, sharp local vibrational mode absorption spectra have been observed for both charge states of the substitutional oxygen defect (Corbett et al., 1961; Bean and Newman, 1971). In Fig. 3-6, the stability of the various vacancy and first generation vacancy-defect pairs is summarized schematically as would be observed in -15 min isochronal annealing studies.

3.2.3 The Silicon Interstitial 3.2.3.1 Interstitial Migration As mentioned in Section 3.2.1, electron irradiation of p-type silicon at 4.2 K produces isolated vacancies and interstitial silicon atoms which are trapped at impurities. The unavoidable conclusion is that interstitial silicon atoms must be highly mobile at 4.2 K at least under the conditions of irradiation in p-type material, and migrate until trapped by impurities. This has been verified directly by EPR for the trapping by

substitutional boron (Watkins, 1975d) and aluminum (Watkins, 1964), and indirectly for gallium (Watkins, 1964). At somewhat higher temperatures (- 100 K), trapping by substitutional carbon also becomes important (Watkins and Brower, 1976). These processes can also be monitored by local vibrational mode (LVM) spectroscopy of the light boron (Tipping and Newman, 1987) and carbon (Chappell and Newman, 1987) atoms, these techniques also revealing similar trapping by interstitial oxygen (Brelot and Charlemagne, 1971). In n-type silicon, the situation is less clear. On the one hand, evidence has been cited that the interstitial silicon atom is more stable, not migrating long distances until -150-175 K (Harris and Watkins, 1984). For example, it is in this temperature region that the carbon interstitials are observed to emerge in n-type material after annealing from a 4.2 K irradiation. Also EPR studies reveal that most divacancies produced at 4.2 K or 20.4 Kin n-type material are perturbed by some other defect nearby. These divacancies disappear in a broad annealing stage beginning at -100 K, consistent with identifying the perturbing defects as the interstitials produced in the pri-

136


mary event and frozen into the lattice nearby, which recombine with their divacancy upon annealing. On the other hand, interstitial boron atoms are observed to be produced at 4.2 K in n-type silicon partially compensated by boron, albeit at an order of magnitude lower rate than in p-type material (Watkins, 1975d). This argues that long range motion is still occurring. Also the emergence of interstitial carbon coincides with the disappearance of two unidentified EPR centers, the annealing kinetics of the dominant one giving (Harris and Watkins, 1984) t - ' = 2 . 8 x 1013exp (-0.57 eVlk,T) s-' (3-4)

The pre-exponential factor is characteristic of a single jump process suggesting that these precursor defects may also be trapped interstitials with the kinetics reflecting the re-release of the interstitials. So, whether significant long range motion of the silicon interstitial is occurring in n-type silicon during electron irradiation at cryogenic temperatures or not is still unclear. We know, however, that the subsequent emergence of interstitial carbon at 150-175 K signals its arrival at the trace substitutional carbon impurities. The 0.57 eV of Eq. (3-4) is therefore an upper limit to its diffusion activation energy in the ntype material. Like the vacancy, therefore, the activation energy for interstitial motion, U o , is small, and the activation energy for its contribution to diffusion, W + U , , must also come primarily from its formation energy, W. Recent local density molecular dynamics calculations determined comparably high mobilities for both the interstitial and the vacancy at -1500 K (Blochl et al., 1993). in agreement with this conclusion. Similarly, recent ab initio local density calculations predict appropriately large formation energies for them, ranging from 3.3 eV (Blochl et al., 1993) to 3.7 eV (Tang et al.,

-

1997). As for the vacancy, this fact has also been difficult to accept, but appears unavoidable.

3.2.3.2 Trapped lnterstitials The structures deduced from EPR studies of some of the trapped interstitial configurations are summarized in Fig. 3-7. In Fig. 3-7 a, the interstitial silicon atom has traded places with a substitutional aluminum atom, ejecting it into the interstitial site. In the A 1 7 state seen by EPR, it resides on center in the empty tetrahedral interstitial site of the lattice (Watkins, 1964; Brower, 1974; Niklas et al., 1985). In Fig. 3-7 b, boron is also displaced from its substitutional site but it prefers to nestle into a bonding configuration between two nearest neighbor substitutional silicon atoms, at least as deduced from EPR studies of its neutral BY state (Watkins, 1975d). The carbon atom, Fig. 3-7c, tends to share a single substitutional site with the extra silicon atom in a split-(100) configuration, as observed by EPR in both its positive and negative charge states (Watkins and Brower, 1976; Song and Watkins, 1990), and by LVM studies in its neutral state (Zheng et al., 1995). Trapped by molecular H,, the structure in Fig. 3-7d has recently been determined from LVM studies (Budde et al., 1998). Here, the presence of the two bonding hydrogen atoms causes the silicon splitinterstitial configuration to rotate to an orientation intermediate between a (100)and (1 10)-split arrangement, as determined from the symmetry and directions of the various hydrogen stretch vibrations. Although observed in this study after hydrogen implantation, these characteristic bands had previously been observed, but not identified at the time, after electron irradiation of silicon grown in a hydrogen atmosphere (Shi et al., 1986). In analogy to V . H,, there-

3.2 Silicon

c.)

b.) BY

a.) A l t f

\ \ \ \ \ \ \ \ \ \ \ \

--

0.13

0

0

0

d.) (SiyHz)'

\- \ \ 0

\ \ \ \ \ \ Ec

0.10

0.37

+

+

0

-

0.17

m++\ \ \ \ \

c:."v-

137

\ \ \ \ \ \

0.28

\ \ \ \ \ \ EV

Figure 3-7. Structures of interstitials trapped by impurities in silicon, and their level positions (eV from the closest band edge). The impurity atoms are shaded (Watkins, 1999).

fore, the defect can be produced when interstitial silicon is trapped by molecular H, in the lattice. Taken together, the configurations of Fig. 3-7 are revealing because they demonstrate the rich variety of configurations available for an interstitial atom. In a sense they provide the most direct experimental clues that we have about what the isolated interstitial silicon atom might look like. Also their electrical and diffusional properties are interesting in this regard, as we now describe. Interstitial aluminum is a double donor and its second donor state E (+/++)hasbeen located at Ev + 0.17 eV (Troxell et al., 1979). Its diffusional activation energy has been determined in p-type material to be 1.2 eV. However, under minority carrier injection, the diffusion is greatly enhanced with a remaining activation energy of only 0.3 eV. Apparently 0.9 eV is being effectively dumped into the defect during an electron-hole recombination event. No experimental information is available concerning the other charge states of the defect

but the arguments of Section 3.2.2.2 suggest large lattice relaxational changes associated with them in order to account for the recombination-enhanced migration. Interstitial boron also displays strong evidence of large lattice relaxational rearrangements with charge state change. Its thermal activation energy for diffusion has been measured to be -0.6 eV but under injection conditions it migrates athermally (Troxell and Watkins, 1980). In addition, it is a negative-U defect, its acceptor level E ( 4 0 ) at -Ec - 0.37 eV being below its single donor level E (O/+) at E, - 0.13 eV (Harris et al., 1987), as shown in Fig. 3-7b. Interstitial carbon on the other hand appears well behaved with an acceptor state at E,0.1 eV and a donor state at E , + 0.28 eV, in normal ordering (Kimerling et a]., 1989; Song and Watkins, 1990). The electrical level structure for the Sii.H2pair has not been determined. However, with all four available bonds of each of the two split silicon atoms satisfied, it is reasonable to expect it to be fully passivated with no levels in the gap.

138


ANNEALING

TEMPERATURE (

OK

)

Figure 3.8, Schematic of the annealing of interstitial-related defects in silicon (- 15 min isochronal) (Watkins, 1999).

Finally, in Fig. 3-8, the stability and evolution of the various EPR and LVM identified interstitial-related defects are summarized schematically, as would be observed in 15 min isochronal annealing studies (Kimerling et al., 1989). When Ci begins to migrate, Ci.C, (Song et al., 1988), C i . O i (Trombetta and Watkins, 1987), and Cigroup-V atom pairs (Zhan and Watkins, 1993) emerge and their stabilities are also indicated. [In addition to these simple Cirelated centers identified by EPR, optical studies have revealed a rich variety of other centers which have been successfully modeled as Ci and Ci-related defects serving as nucleation centers for interstitial silicon (Davies, 1988: Davies et al., 1987)l. Similarly, when Al, migrates, it forms Al, . Al,. pairs (Watkins, 1964; Niklas et al., 1985). The fate of Bi has not been established by EPR or LVM. However, DLTS studies have been interpreted to indicate

-

(not shown in Fig. 3-8) that, like Ali, it migrates to form electrically active complexes with substitutional boron and also with substitutional carbon, as well as interstitial oxygen (Kimerling et al., 1989). Also not shown is indirect DLTS evidence of the trapping of interstitial silicon by interstitial oxygen and its re-release at -250 K (Harris and Watkins, 1984).

3.2.3.3 Theory of the Silicon Interstitial In the absence of direct experimental observation of the isolated interstitial, we turn to theory. Several independent groups have modeled the silicon interstitial using modern state-of-the-art ab initio local density quantum mechanical calculations, and the results are remarkably consistent with our earlier clues. In Fig. 3-9, the results of two of them (Car et al., 1985; Bar-Yam and Joannopoulos, 1984, 1985) are combined

3.2 Silicon

139

Figure 3-9. Total energy (CC) diagram and stable configurations predicted from theory (Car et al., 1985; BarYam and Joannopoulos, 1984, 1985) for three ionization states of interstitial silicon. The energy level positions predicted from these results are illustrated on the right (Watkins, 1997).

into a CC diagram. We note that the predicted stable configurations are different for the three charge states: Si;+, as AI++in Fig. 3-7a, in the high symmetry tetrahedral position (T),Sit, as BY in Fig. 3-7b, in the bondcentered configuration ( B ) , and Si:, somewhat like Ci in Fig. 3-7c, or S i i . H2 in Fig. 3-7d, in a (100) split configuration (X). Further, since each of these configurations represent saddle points for migration between any of the others, the indicated energies can provide estimates of the activation energy for migration in each charge state. We note that the predicted barrier is particularly small in the Si: charge state, perhaps already providing an explanation of the high observed mobility of the interstitial. But, in addition, we are immediately led to another mechanism, when we recognize that these results imply that simply changing its charge state by the capture of a carrier invariably brings it to a saddle point configuration from which further carrier

capture or release allows migration. Recognizing that there is significant ionization produced during electron irradiation, this athermal, recombination-enhanced process provides a logical explanation for the high mobility observed in p-type studies. Also implied in the curves is the remarkable level position ordering indicated to the right of the figure. The true electrical level position of a defect is defined as the energy difference between the relaxed ionized and un-ionized states. The first donor level E (O/+) position below the conduction band edge is therefore given by the energy difference between Sip in its lowest energy X configuration and Si: + e- in its lowest energy B configuration, predicting E, - 1.2 eV, i.e., right at the valence band edge! The second donor level is given, in turn, by the energy difference between the B configuration of Sit + e- and the T position of Sit' + 2e-, giving E,-0.4 eV. A very large negative U ordering is therefore predicted,

140


perhaps explaining why the paramagnetic Si; charge state has not revealed itself even in n-type material, where the interstitial appears to be stable at least up to -140 K. In such a negative U situation, the intermediate Si; EPR-active charge state is thermodynamically unstable, and with the neutral state so deep it may be difficult to optically excite it, as was possible in the vacancy and interstitial boron, negative U cases. One question that must be addressed is how accurate are such calculations? Can we trust them? At present, the consensus seems to be that they may be reasonably reliable much better in fact than there is good theoretical basis to expect. For example, they have proven to be remarkably successful in the case of some of the trapped interstitial configurations, where the predicted configurations can be compared directly to those that have been experimentally established: The experimentally determined configuration in Fig. 3-7d for Sii.H2 is found to agree in striking detail with recent theoretical predictions (Van de Walle and Neugebauer, 1995; Budde et al., 1998). Similarly, recent theoretical calculations have confirmed the experimentally observed configurations for interstitial carbon (Capaz et al., 1998; Leary et al., 1997). In the case of interstitial boron, a configuration similar to that observed by EPR has been predicted for the neutral state, and also for the negative one (Tarnow, 199 1 ). However, the lowest energy configuration for the positive charge state i n the calculations was found to occur as the boron atom reassumes its substitutional position, ejecting the silicon atom back into the nearby tetrahedral interstitial site. These predicted configurational changes fit in quite nicely with several otherwise difficult to explain EPR observations, and are currently believed therefore to be essentially correct. They also provide a possible explanation for the negative U ordering of its

energy levels, and its experimentally observed, recombination-enhanced migration. Caution must be exercised, however, in over-stating their accuracy. Calculations by others for the silicon interstitial, obtain qualitative but not necessarily detailed quantitative agreement with the results shown in Fig. 3-9. In one (Chadi, 1992), for example, the various configurations shown in the figure are also found to be close in total energy, but with the (1 10) split configuration the lowest for all charge states. Also, the electrical level positions estimated by such local density calculations, and as indicated in Fig. 3-9, may have a considerable further uncertainty because they result from inserting the experimental silicon bandgap, while the calculations give only one half that value. Nevertheless, it is clear that the calculations are coming up with qualitatively reasonable results which can explain the high mobility of the silicon interstitial and its configurations and properties as evidenced in its trapped states. In spite of the fact therefore that the isolated interstitial silicon atom has not been directly observed, we have reason to feel we may understand it pretty well. Like the vacancy, the evidence is that it diffuses easily through the lattice, having migrated large distances in n-type silicon by -140 K and, under the electron irradiation conditions in p-type, very efficiently, even at 4.2 K. 3.2.4 Close Frenkel Pairs

It has commonly been assumed that recombination annihilation of the close interstitial-vacancy pairs, process (a) in Fig. 3-1, is also occurring in the same cryogenic temperature range where the interstitial becomes mobile. No evidence of the pairs has been apparent in either EPR or electrical studies. Recent X-ray diffraction studies

3.3 Other Group IV Semiconductors

have suggested, however, that this may not be correct (Ehrhart and Zillgen, 1997). In these studies, the persistence of diffuse Xray scattering produced by electron irradiation upon annealing to above room temperature has been interpreted as resulting from highly stable, close pairs which remain. This, of course, could possibly be true; the result of a large, final barrier for annihilation. A recent tight binding simulation has indeed predicted such a barrier (Tang et al., 1997). It will be interesting to see if this turns out to be correct or not. Either way, however, it has no direct bearing on the understanding we have obtained concerning the escaped, isolated vacancies and interstitials.

3.3 Other Group IV Semiconductors 3.3.1 Germanium Much less has been established for germanium. A defect identified as a vacancy trapped by interstitial oxygen with a configuration similar to the silicon counterpart (Fig. 3-5 d) has been observed by both EPR (Baldwin, 1965) and LVM (Whan, 1965). It emerges at 130 K, suggesting easy vacancy migration, as in silicon. Interstitial germanium complexed with two hydrogen atoms has also been identified by LVM studies (Budde et al., 1998), and its configuration is almost identical to that for the corresponding Sii . H, complex (Fig. 3-7d). From these limited observations, we may tentatively conclude that the properties of the intrinsic defects in germanium are closely similar to those in silicon.

-

3.3.2 Diamond The isolated vacancy has been observed in its negatively charged state, V-, by EPR

141

and ENDOR (Isoya et al., 1992). As for the vacancy in silicon, the wave function is highly localized in the dangling bonds of the four nearest carbon neighbors. However, unlike the vacancy in silicon, it has spin S = 3/2, and is undistorted with full T, symmetry. Here, apparently, the electron-electron Coulomb interactions dominate over the Jahn-Teller coupling, and Hund’s rules therefore apply, and the electrons go, one each with spin up, into the three degenerate t2 orbitals of Fig. 3-2, producing a nondegenerate 4A, closed a:t: shell and no distortion. We will see that this pattern becomes more evident on going to the higher bandgap materials, which tend to produce more highly localized states for the electrons. An optical absorption spectrum, with a zero phonon line at 3.150 eV and labeled ND1, has also been identified with V - (Davies, 1994). The neutral vacancy has been identified from detailed study over many years of its GRl optical absorption and luminescence band, with a zero phonon line at 1.673 eV (Davies, 1994). As for the neutral vacancy in silicon, its ground state has spin zero as predicted from Fig 3-2 for the a:ti configuration, and it undergoes a tetragonal Jahn-Teller distortion. Unlike the vacancy in silicon, however, the distortion is not static but dynamic, a result of the much stronger restoring forces in diamond. Also, although the many-electron effects are therefore not immediately apparent in its ground state, they are, however, very important in understanding the excited states. In particular, the excited state to which the GRl transition occurs also derives primarily from the a:t: configuration, the energy difference from the ground state coming therefore primarily from the many-electron interactions. EPR has been observed for the long-lived 5A, excited state of the neutral vacancy (derived from the a; t; configura-

142


tion) under optical excitation (van Wyk et al., 1995). No optical transitions have been identified for the positive vacancy, but an EPR signal has been tentatively identified for it in p-type material (Isoya et al., 1994). As predicted for its aTtl configuration (Fig. 3-2), it has S = 1/2 and is statically (100) distorted. Assuming that it is indeed the isolated vacancy, the distortion would reflect a Jahn-Teller distortion, as for silicon. The activation barrier for neutral vacancy migration has been extracted from annealing studies of the GRI band to be 2.3 & 0.3 eV (Davies et al., 1992). The vacancy can be trapped by substitutional nitrogen, forming a V . N pair, which has been observed in its negative charge state by EPR (Loubser and van Wyk, 1977), optical spectroscopy (duPreez, 1965; Davies and Hamer, 1976), and a variety of optical detection methods of EPR and ENDOR (van Oort et al., 1988; Redman et al., 1991; He et al., 1993; Lenef et al., 1996). Like the vacancy trapped by Group V donors in silicon, the nitrogen donor is a nearest neighbor of the vacancy in diamond. However, unlike the S = 0 nonparamagnetic silicon case, the ground state is the Hund's rule S = 1 nondegenerate state and does not therefore Jahn-Teller distort by pairs, as shown in Fig. 3-5e. Observation of the neutral divacancy by EPR has also been observed and confirmed from resolved 13C hyperfine interactions in an isotope-enriched, synthetic diamond crystal (Twitchen et al., 1999). In this case, the ground state has S = 1, but a Jahn-Teller distortion still occurs similar to that for the silicon divacancy, but of opposite sign. The isolated carbon interstitial has not been observed, but an EPR center identified as an interstitial pair in a di-(lOO)-split configuration has recently been reported

(Twitchen et al., 1996). It can be formed by high-energy electron irradiation at room temperature, indicating that the interstitial is mobile at or below that temperature. [Note added in proof, 12/99: EPR of the isolated interstitial has now been reported (Hunt et al., 1999), revealing a split-(100) configuration for it, similar to its configuration in silicon, Fig. 3-7c. Evidence for its migration during the irradiation is also cited.]

3.3.3 Sic The negatively charged isolated silicon vacancy, Vii, has been identified by EPR in each of the common 3 C (Itoh et a]., 1990, 1997), 4H (Wimbauer et al., 1997a), and 6 H (Schneider and Maier, 1993) polytypes. Again, the molecular orbital model developed for the silicon vacancy in Fig. 3-2 provides a useful starting point, giving three electrons in the t2 orbital for that charge state. As in diamond, these orbitals are highly localized on the four nearest carbon atoms, forcing the Hund's rule, undistorted S = 3/2 ground state, which has been directly confirmed from ENDOR studies on the 4H polytype (Wimbauer et al., 1997a). ODEPR studies have also been reported for the neutral silicon vacancy in the 4 H and 6H polytypes, as observed in photo-luminescence (Sorman et al., 1998). In those experiments, the excited emitting state observed has S = 1. The vacancy disappears upon annealing at -750 " C , but no information is available concerning its electrical level structure or interactions with other defects. An EPR signal identified with the positively charged C vacancy, VG has also been reported for 3C-Sic (Itoh et al., 1992, 1997). As expected for its ait; configuration, it has S = %, and displays what is presumed to be a Jahn-Teller distortion along

143

3.4 I l - V I Semiconductors

a (1 00) direction. However, the symmetry of the observed distortion, D,, is somewhat unexpected for a Jahn-Teller distortion (D2d expected, as in silicon), and the 29Si hyperfine interaction on the four equivalent nearest neighbor silicon atoms accounts for only -30% of the electron spin, indicating an unexpectedly diffuse t, wave function. We will therefore continue in this review to consider this identification as tentative only. No information is available for either of the two interstitials.

p

(0,)

NC

3.4 11-VI Semiconductors 3.4.1 ZnSe 3.4.1.1 The Zinc Vacancy Electron irradiation (1-3 MeV) at or below room temperature produces isolated zinc vacancies which have been studied in considerable detail by EPR (Watkins, 1971, 1975e, f, 1977) and ODEPR (Lee, 1983; Lee et ai., 1980, 1981; Jeon, 1988; Jeon et al., 1986; Watkins, 1990; Jeon et al., 1993). The zinc vacancy produces a double acceptor E(=/-) in the lower half of the ZnSe band gap, and in normally n-type material it is doubly negatively charged (VZ=,)and is not paramagnetic. Illumination with 500 nm light serves to photoionize it producing Vgn which is observed by EPR. The structure of V;, deduced from these studies is shown in Fig. 3-loa. The hole (unpaired spin) is highly localized on only one of the four nearest Se neighbors and is highly p-like pointing toward the vacancy. This "small polaron" structure follows naturally from a simple one-electron molecular orbital model for the defect, similar to that of Fig. 3-2 developed for the silicon vacancy. As for the silicon vacancy, the a, and t2 orbitals in the undistorted T, symmetry are made up as linear combinations of

-

Td

-

c3v

I

A1

Figure 3-10. a) Simple one-electron molecular orbital model for V,, in ZnSe. b) The corresponding many-electron states (Watkins, 1990).

the p-orbitals pointing into the vacancy on each of the four Se (a, b, c, d) neighbors. For V;,, its eight electrons completely fill the orbitals, no degeneracy remains, the manyelectron sit: state is of A, symmetry, and no Jahn-Teller distortion occurs. Vg,, however, has degeneracy associated with a missing electron in the t2 orbital, is of T, symmetry, and is therefore unstable to a Jahn-Teller distortion, which in this case is trigonal, as shown. In Fig. 3-10b, the

144


energies of the corresponding ground and excited many electron states (sum of the one-electron energies of Fig. 3-loa) are also shown. Also shown are optical absorption transitions predicted by the model. These have been observed experimentally as broad absorption bands centered at 1.4 eV (n, polarized parallel to the trigonal distortion axis) and at 1 .O eV (0, polarized perpendicular to the axis) and confirmed unambiguously to arise from the zinc vacancy by ODEPR studies. The presence of the zinc vacancy also produces luminescence at 1.72 eV (720 nm) which has been established by the ODEPR studies to arise from electron transfer from a distant shallow donor to the E (=/-) acceptor level of the vacancy. Combining all of this optical information, a CC diagram has been constructed for the double acceptor state of the defect. This is shown in Fig. 3-1 1, where Q is the amplitude of the trigonal distortion. The lowest energy curve is for the V,, charge state which is stable in the undistorted position, Q = 0. Shown also at E, = 2.82 eV higher in energy is the curve for V;n plus a free electron and hole. The remaining curves are for the ground and excited states of V:, + e-, as given in Fig. 3- 10b. (The CC diagram is for the total energy, the curvature upward for each state reflecting the elastic energy stored in the lattice due to the distortion, which must be added to the electronic energies of Fig. 3-lob.) The curves shown in Fig. 3-1 1 for each state have been derived from a simple linear Jahn-Teller coupling theory with a single elastic restoring force constant for all states (Schirmer and Schnadt, 1976). They are completely determined by matching to the four observed optical transitions - the 1 .O eV and 1.4 eV absorption bands of VS,, the donor-acceptor pair luminescence at 1.72 eV, and the photoionization and lumi-

( A i l Vin+ e €,T=

0.35 eV

L1 = 0.43 eV

EA = 0.66 eV

EG=2.82eV

6 .e 0

._

1

u X

W

i Figure 3-11. Configuration coordinate model for the second acceptor state E(=/-) of V,, in ZnSe (Watkins, 1990). e , denotes an electron trapped on a shallow donor.

nescence excitation band at 2.51 eV. That this is a good representation of the defect is confirmed by other experimental observations: (1) The splitting of the T, level vs. Q matches well an estimate of the Jahn-Teller coupling coefficient from uniaxial stress studies of the VS, spectrum. (2) Figure 3-1 1 predicts the double acceptor level E (=/-) to be at Ev + 0.66 eV (the energy difference between V& + e- and V & + e- + hf, each determined in its relaxed state). This agrees with a value of E, + 0.7 eV estimated from the kinetics of hole release from VZ, observed by EPR. The CC diagram also contains other important information: It reveals the magnitude of the Jahn-Teller energy to be 0.35 eV, the A,-T, "crystal-field'' splitting to be 0.43 eV, and the characteristic phonon frequency associated with the trigonal distortion to be -1 1 meV (from the curvature of the V,, curve), a reasonable value, being of the order of characteristic TA phonons in ZnSe. The zinc vacancy in ZnSe is therefore an extremely well characterized center as far

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3.4 Il-VI Semiconductors

as its electrical and optical properties and its microscopic electronic and lattice structures are concerned.

being lowered by the presence of the impurity. The donor-vacancy pairs have also been established to be important defects in asgrown material giving rise to the characteristic orange-yellow "self-activated" luminescence in ZnSe (Nicholls et al., 1978; Lee et al., 1980, 1981).

3.4.1.2 Zinc Vacancy Diffusion and Interaction with Other Defects Figure 3-12 shows schematically the result of annealing after a 4.2 K or 20.4 K electron irradiation (Watkins, 1977). The vacancy is stable up to -400 K at which point it disappears and new defects emerge that have been identified as vacancy-impurity pairs. This establishes that the annealing is a result of long range migration of the vacancy to be trapped by the impurities. The kinetics of the process have been studied giving for the zinc vacancy migration barrier, U , = 1.26 eV

3.4.1.3 Interstitial Zinc and Close Frenkel Pairs In addition to the zinc vacancy, zinc interstitials and several distinct zinc-vacancy-zinc-interstitial close Frenkel pairs have been observed to be frozen into the lattice after 1-3 MeV electron irradiation at T S 20.4 K. In this case, as opposed to silicon, the expected scenario of Fig. 3-1 is indeed observed. Four distinct close pairs have been observed directly by EPR (Watkins, 1974, 1975e, f; 1977, 1990). Labeled I through IV, they anneal in discrete steps between 60 K and 180 K as illustrated in Fig. 3-12. Their electronic structure, deduced from the EPR studies, remains, like that of the vacancyimpurity pairs above, essentially that of a VZ, perturbed only slightly by the presence of interstitial Zn:+ in its different nearby lattice positions. In addition, many more

(3-5)

The zinc vacancy-impurity pairs that have been identified include zinc vacancies adjacent to the isoelectronic impurities S and Te, and to substitutional chemical donors (Holton et al., 1966). In all cases, the electronic structure is very similar to that of the isolated zinc vacancy, the hole again highly localized on one of the four vacancy near neighbors but with the symmetry of the center

0

100

200 300 LOO Annealing temperature [ K ]

145

500

600

Figure 3-12. Schematic of the annealing stages (15 min isochronal) for the Vznrelated defects observed by EPR after 1.5 MeV electron irradiation at 20.4 or 4.2 K (Watkins, 1990).

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146


distinct close pairs, as well as the isolated interstitial zinc atom, have been detected by ODEPR in luminescence bands in the visible and near infrared produced by the irradiation (Rong and Watkins, 1986a, b, 1987; Rong et al., 1988; Watkins et al., 1988; Rong et al., 1996: Barry and Watkins, 1996). Well resolved spectra for at least 25 distinguishable close pairs (different separation distances) have been observed. They anneal in a series of discrete steps, the closest ones first, the more distant ones at progressively higher temperatures, with the isolated interstitial zinc last at -260 K, as indicated also in the figure. No significant increase in the isolated vacancy EPR intensity is observed through the annealing stages indicating that the primary annealing process for the pairs is annihilation due to recombination (process (a) in Fig. 3-1). In these ODEPR studies, the interstitial is observed in its Zn: paramagnetic state, either when isolated, or exchange-coupled with its paramagnetic Vsn partner in the close pairs. From a scientific viewpoint, the pairs are particularly interesting because they provide a unique system where the exchange and Coulomb interactions, the radiative lifetimes, and the overall combined wavefunctions can be probed directly vs lattice separation. The interested reader is encouraged to refer to the original literature. Here we will concentrate only on what has been learned about the isolated interstitial. Two distinct configurations for the isolated Znc have been identified (Rong et al., 1996; Chow and Watkins, 1998): one in the tetrahedral site surrounded by four selenium atoms, (Zr&, the other in the tetrahedral site surrounded by four zinc atoms, (Zni')zn. In each case, hyperfine interactions have been resolved for the central zinc and one or more shells of neighboring selenium. The results establish that in each case

the Zn? ion is on-center in the tetrahedral site, and its electronic structure is that of an electron in a highly localized S-orbital bound to the Znt+ closed shell core. By analysis of the luminescence wavelength shift versus the close pair distance, and the Bohr radius deduced from hyperfine analysis for isolated (Znf),,, it has been possible to estimate the level position for its second donor state E (+/++) as -E, - 0.9 eV. From analysis of the (Zn;),, hyperfine interactions, its second donor level is estimated to be -0.5-0.7 eV deeper. This agrees well with earlier theoretical predictions (Laks et al., 1992; Van de Walle and Blochl, 1993). Detailed kinetic studies of the annealing stages involving the interstitial in Fig. 3-12 have not been performed. However, assuming a characteristic single jump rate Y

- 1013exp (- U,/kBT)

(3-6)

and 102-105 jumps required before being trapped or annihilated in 15 min at T 260 K for the distant pairs and for isolated zinc leads to an estimate for the interstitial migration barrier of

U,

- 0.60 - 0.70 eV

(3-7)

Migration of the interstitial also occurs under electronic excitation at 1.5 K. This athermal, recombination-enhanced migration has been demonstrated indirectly by observing annealing of the different pairs, and directly by the conversion between the two interstitial configurations, the onejump process, under optical excitation at 1.5 K (Chow and Watkins, 1998). 3.4.1.4 Defects on the Selenium Sublattice An EPR center identified with the selenium vacancy, Vg,, has been reported by Gorn et al. (1990). It is isotropic, similar to F-centers in the ionic NaCl crystal structure lat-

147

3.4 I l - V I Semiconductors

tices (Foin the alkali halides, F+in the alkaline earth oxides), with an unpaired electron highly localized but symmetrically distributed primarily over the four nearest neighbors of the vacancy. Weak resolved hyperfine interactions with the twelve next neighbor "Se atoms aided in the identification.

Table 3-2. Level positions and migration energies (in eV), determined for intrinsic defects on the metal sublattice in ZnSe and ZnS. ZnSe

ZnS ~

~

~~

Vzn

E (4-)

E,+0.66

E,+ 1.1

ua € (+/++) E (+/++)

Zn,

ua

1.26 -€c - 0.9 -Ec - 1.5 0.6 - 0.7

1.04

(Zn,),, (Zn,),,

3.4.2 Other 11-VI Materials 3.4.2.1 The Metal Vacancy The metal vacancy has also been observed by EPR in its single-negative charge state in ZnS (Watkins, 1975f, Shono, 1979), CdS (Taylor et al., 1971), CdTe (Emanuelsson et al., 1993), B e 0 (HervC and Maffeo, 1970; Maffeo et al., 1972; Maffeo and HervC, 1976), and ZnO (Taylor et al., 1970; Galland and HervC, 1970, 1974). The electronic structure in each case is very similar to that in ZnSe, the hole being highly localized in a p-orbital on a single chalcogen neighbor. In the case of ZnS, correlative optical (Watkins, 1973) and ODEPR (Lee et al., 1982) studies have identified an optical absorption band at 3.26 eV associated with Vg,, one at 1.65 eV associated with V?,, and shallow donor to Vz, recombination luminescence at 2.18 eV. From this, a CC diagram could again be constructed, similar to that in Fig. 3-1 1 for the zinc vacancy in ZnSe. It predicts that for V,, in ZnS, the double acceptor level E(=/-) is at -Ev + 1.1 eV, and the Jahn-Teller relaxation energy is -0.5 eV. Annealing kinetic studies have been performed for the disappearance of the VZn spectrum in ZnS and give an activation energy of 1.04 ? 0.07 eV (Watkins, 1977) (see Table 3-2). The pre-exponential factor for the decay is consistent with long range migration. The activation energy has there-

fore been tentatively identified with the migration activation energy for the vacancy. It must be considered tentative, however, because vacancy-impurity pairs were not observed to grow in during the process. Vacancy-donor pairs have been observed in as-grown material and, again, their structures are very similar to those of the corresponding pairs in ZnSe (Holton et al., 1966). In the case of CdS, CdTe, BeO, and ZnO, no optical or annealing information is available, except that the vacancies are stable at 300 K. An interesting additional bit of information does come, however, from studies in B e 0 (Maffeo et al., 1970) and ZnO (Galland and HervC, 1970). There, a deep neutral state is also stable in the gap and has been observed by EPR. The defect has an S = 1 ground state, the electronic structure being that of two bound holes, one on each of two neighbors. Again, in terms of our simple one-electron molecular orbital model (Fig. 3-10), this can only occur if the many-electron effects are greater than the Jahn-Teller-produced crystal-field energies, forcing Hund's rules (maximum spin). And so, at least for the wider bandgap oxides, many-electron effects can be important in the 11-VI materials, as well as in S i c and diamond.

148


3.4.2.2 The Chalcogen Vacancy The chalcogen vacancy has been observed by EPR and ENDOR in its singly positive charge state i n ZnS (Schneider and Rauber, 1967), CdTe (Hoffmann et al., 1994), ZnO (Smith and Vehse, 1970; Gonzalez et al., 1975), and Be0 (DuVarney et al., 1969; Garrison and DuVarney, 1973). As for ZnSe, the centers are similar to Fcenters in the ionic NaCl crystal structure lattices, (Fo in the alkali halides, F+ in the alkaline earth oxides), with an unpaired electron highly localized but symmetrically distributed primarily over the four nearest neighbors of the vacancy. Optical transitions have been associated with the chalcogen vacancies in some cases, indicating that the second donor levels E (+/++) are deep, being 2-3 eV below the conduction band. Annealing studies of the optical bands associated with the sulfur vacancy i n ZnS have led to an estimate of its migration activation energy of 1.4 eV (Matsuura et al., 1987). No other systematic studies of the migrational properties have been reported.

3.4.2.3 The Other Intrinsic Defects Interstitial metal and chalcogen atoms are also produced in a radiation damage event, and must therefore be present in isolated or trapped form. ODEPR studies of ZnTe after 2.5 MeV electron irradiation in situ at 4.2 K have been interpreted as detecting close Frenkel pairs on the tellurium sublattice (Ittermann et al., 1994; Uftring, 1998). From analysis of the symmetry and Zeeman splittings of the pair S = 1 spectra, it was tentatively concluded that the tellurium interstitial occupies a low symmetry bonded configuration in the lattice. Annihilation of the spectra under thermal (-30120 K ) and/or optical injection (4.2 K), revealed that, like interstitial zinc in ZnSe,

it too has a low barrier for thermal migration and can migrate athermally under electronic excitation. This represents the only available information on interstitials in the 11-VIS, other than that discussed earlier for ZnSe. No evidence exists for antisites (atom A on site B, etc.) in the 11-VI materials. This is as expected considering the large chemical difference between the group I1 and group VI constituents.

3.4.3 Theory There have been several recent theoretical studies using the modern state-of-the-art local density techniques to probe the properties of the intrinsic defects in the 11-VIS. Already mentioned is the good agreement with experiment for Zni in ZnSe (Laks et al., 1992; Van de Walle and Blochl, 1993). A number of unexpected, and therefore very challenging, predictions have also been made, however. In particular, Chadi (1997) has predicted that vacancies on either sublattice may be amphoteric, acting as both double donors and double acceptors. For example, the cation vacancy may not only introduce double acceptor states, as established experimentally, but may also take on the +2 charge state in p-type material, as an anion atom neighbor moves into the vacancy, producing an anion antisite (anion on the cation site) with a neighboring anion vacancy. Similarly, the anion vacancy may not only introduce a double donor level, as experimentally established, but it may also trap additional electrons to become -2 charged; in this case, the result of dimerization of the metal neighbors. Also, for the anion vacancy in ZnSe, large breathing relaxations have been predicted to cause the first donor level E (O/+) to be below the second donor level E (+/++) in negative-U ordering (Garcia and Northrup, 1995), mak-

3.5 I l l - V Semiconductors

ing the +I charge state studied by EPR thermodynamically unstable. No experimental evidence of any of these properties exists at present, but the predictions cannot be summarily dismissed without further careful experimental studies.

3.5 111- V Semiconductors 3.5.1 Antisites In the 111-V semiconductors, a class of intrinsic defects not considered for the elemental group-IV and the compound 11-VI materials becomes important. It is the antisite, where a group-V atom occupies a group-I11 atom site (denoted VI,,) or vice versa (111”). The first, the anion antisite, should, in the usual chemical impurity sense, be a double donor. The second, the cation antisite, should be a double acceptor. These become important because the chemical difference between the group I11 and V atoms is sufficiently small that each has a finite solubility on the other sublattice. They are therefore present in as-grown materials. They can also be produced by electron irradiation and by plastic deformation.

3.5.1.1 The Anion Antisite VrII The presence of VI,, antisites has been established directly by EPR, ODEPR, and ODENDOR in GaP (PGa.)(Kaufmann et al., 1976, 1981; O’Donnell et al., 1982; Killoran et a]., 1982; Lappe et al., 1988), GaAs (AsGa) (Weber and Omling, 1985; Meyer et al., 1987; Spaeth and Krambrock, 1994), and InP (Pin) (Kennedy and Wilsey, 1984; Deiri et al., 1984; Kanaah et al., 1985; Jeon et al., 1987). As expected they appear to be deep double donors, the magnetic resonance studies being performed on the singly positively charged paramagnetic

149

(S= 1/2) state. In this state, the group-V atom is located on-center in the group-I11 atom site and the paramagnetic electron is strongly localized in a Coulombically bound S-like orbital (a,) on the atom. However, because the defects are being observed in an S-state, the ODEPR and EPR spectra are relatively insensitive to whether the defect is truly isolated or whether it is complexed with some other defect nearby. In Gap, the ODEPR and EPR resolution is sufficient to resolve hyperfine interactions with the nearest neighbor phosphorus atoms. In this case two different antisiterelated centers can be distinguished in both EPR and ODEPR studies, one with four P neighbors (PP,), as expected for an isolated antisite, and one with only three [PP3X in EPR (Kennedy and Wilsey, 1978), PP3Y in ODEPR (O’Donnell et al., 1982; Killoran et al., 1982; Rong et al., 1991; Sun et al., 1994), which may or may not be the same defect]. The latter presumably have an impurity or a vacancy replacing one of the P nearest neighbors but otherwise no firm identification is yet available. But even in the case of PP,, higher resolution ODENDOR studies have reported that a distinguishably different, antisite-related defect also appears to have four equivalent nearest phosphorus atoms and twelve equivalent next nearest gallium neighbors, becoming a second candidate for the isolated antisite (Lappe et al., 1988). Perhaps neither is. In the case of InP, slight shifts in the EPR spectra between n- and p-type materials provided an early hint also of such effects which have been confirmed subsequently in ODENDOR studies (Sun et al., 1993). There, several distinct PI, ODENDOR spectra have been revealed, one of which appears to be consistent with an isolated antisite, but the others are not. In GaAs, evidence that more than one antisite-related defect may be involved comes from EPR

150


studies where the spin-lattice relaxation times differ greatly for as-grown defects vs those produced by plastic deformation or radiation damage (Hoinkis and Weber. 1989). ODENDOR studies also indicate more than one (Spaeth and Krambrock, 1994). This points up the extreme difficulty that may occur in making this determination from the magnetic resonance study of an Sstate system. In effect, the presence of a nearby defect often makes little difference to the electronic structure of the antisite. On the other hand, it is of extreme importance in understanding the electrical and optical properties of the total defect, its stability, etc. An interesting and much studied example of a VI,, antisite-related defect with this ambiguity is a center labeled EL2 in GaAs (Martin and Markram-Ebeid, 1986; Baraff, 1992). The importance of this native defect is the fact that, being deep, it serves to make semi-insulating GaAs, desirable for device substrates. The scientifically intriguing feature of the defect is that it displays metastability. This is illustrated by the CC diagram of Fig. 3- 13a. In its stable configuration (A), it is a deep donor at E, - 0.75 eV which can be detected by DLTS and by an optical absorption band at 1.15 eV. Optical excitation into this band converts the defect into a metastable configuration (B) for which no optical absorption is observed, the vertical optical ionizing transition now being above the band gap, as shown. Conversion back to the stable configuration occurs with an activation energy barrier of -0.3 eV. EPR, ODEPR, and ODENDOR studies have provided convincing evidence that EL2 involves the AsGa antisite (Weber and Omling, 1985; Meyer et al., 1987). Theoretical studies have provided strong arguments that EL2 arises from the isolated antisite

(Dabrowski and Scheffler, 1989; Chadi and Chang, 1988; Kaxiras and Pandey, 1989). They predict that the metastable state can result from the ejection of the AsGa atom into an interstitial position leaving a gallium vacancy behind (Fig. 3-1 3 b). The energetics that are calculated agree closely with the experimental observation for EL2. Experimentally, optical studies of a zerophonon line at 1.04 eV, believed to be associated with the EL2 1.15 eV photo-excitation band, have provided strong evidence that the stable state of the center indeed has full tetrahedral symmetry (Kaminska et al., 1985; Trautman et al., 1990). In addition, study of the EL2 conversion between its two configurations under uniaxial stress and polarized light excitation by monitoring its broad 1.15 eV band has provided strong evidence that the metastable state has trigonal (1 11) symmetry and the stable state full undistorted tetrahedral symmetry (Trautman and Baranowski, 1992). These conclusions have been further supported by electrical measurements under uniaxial stress (Babinski et al., 1994). Taken together, they give strong confirmation that metastability is an intrinsic property of the isolated AsGa

b)

d2 As

Ga

A

Figure 3-13. a ) CC diagram for EL2 in GaAs (energies in eV). b) The presently accepted model as that of an isolated AsGa antisite which is ejected into a metastable adjacent interstitial position.


antisite, and with configurational changes similar to those predicted by theory. On the other hand, ODENDOR results have been interpreted to indicate that the defect is actually an AsG~-As~ pair, the metastability presumably arising from the interstitial arsenic which hops from one separation distance to another (Von Bardeleben et al., 1986; Spaeth and Krambrock, 1994). As a result, there has been much controversy in the literature over the years concerning the correct structure of the EL2 defect. Very recently, however, a careful comparison of the ODENDOR results with the experimentally observed line-shape of the EL2 ODEPR spectrum has led to the conclusion that the extra arsenic signal seen in the ODEPR, and thought to be coming from a nearby interstitial, cannot be part of EL2. It must be coming from some other defect whose optical absorption overlaps that of EL2 (Wirbeleit and Niklas, 1997). This may have finally settled the controversy, the weight of the evidence now strongly favoring identification with the isolated AsGa antisite, as indicated in the figure. In some cases, the perturbation of a nearby defect has been sufficient to detect directly in EPR or ODEPR. We have already mentioned PP,X and PP3Y in Gap. Additional PG,-defect complexes have been detected by these techniques in electron-irradiated (Kennedy and Wilsey, 1981) and impurity-diffused (Godlewski et al., 1989; Chen et al., 1989) Gap. In InP, a lower symmetry PI,-related center has also been detected by ODEPR (Kennedy et al., 1986). In the case of GaAs, an EPR center could be identified as a perturbed AsGaantisite from its weaker central hyperfine interaction. Labeled AsGa-X2, it has been subsequently tentatively identified from ODENDOR as an AsGa antisite with an arsenic vacancy as its nearest neighbor (Koschnick et al., 1998).

151

3.5.1.2 The Cation Antisite 111, No direct magnetic resonance identification of a 111, antisite exists at present for any 111-V semiconductor. On the other hand, an impurity group-I11 atom cation antisite - boron on the As site in GaAs - has been identified by local vibrational mode (LVM) spectroscopy (Gledhill et al., 1984). In this case, resolved structure due to the two gallium isotopes for each of the four neighbors is observed providing a direct confirmation. This provides strong evidence that the intrinsic IIIv antisite should also be present and therefore play an important role in the properties of the material. In Ga-rich GaAs, a photoluminescence peak at 1.441 eV and an electrical acceptor level at Ev + 0.078 eV have been suggested to arise from the GaAs antisite (Yu, 1983; Elliot et al., 1982; Roos et al., 1989). This appears to be a reasonable deduction, confirmed somewhat by the fact that BAS also appears to have an acceptor level near the valence band as indicated by the observation that the local mode disappears abruptly when the Fermi level goes below E , + 0.2 eV (Woodhead et al., 1983). Theory has again made some interesting predictions for the 111, antisite. In the first place, in one set of calculations, it has been predicted to be the dominant defect in Garich GaAs, and when on-center in the As site, to be a double acceptor with its first acceptor level E ( 4 0 ) at E, + 0.25 eV (Baraff and Schluter, 1985). A more recent calculation has led to the interesting conclusion that the Ga atom is on-center only in the Gal, state. Upon two hole capture, it is predicted to break its bond with one Ga neighbor and eject toward the interstitial position, forming what can be considered a closely bound Gai+ VAS pair (Zhang and Chadi, 1990). It is predicted to be a negative-U center with the E (=/O) occupation

-

152


level at -E, + 0.17 eV. (The coincidence of this level position with that for which the LVM BASband disappears is suggestive, as pointed out by these authors.) In conclusion, we have in GaAs some tentative experimental indications for the properties of the intrinsic 111" antisite and an intriguing suggestion of structural instabilities for it from theory. The lack of magnetic resonance identification in this or any of the III-V materials, however, still makes this only speculation at present.

3.5.2 Group-I11 Atom Vacancies The group-I11 atom vacancy has been identified only in Gap. The identification came first from EPR studies of room temperature electron-irradiated GaP where the characteristic hyperfine structure of the four phosphorus neighbors is resolved (Kennedy and Wilsey, 1981; Kennedy et al., 1983). Subsequent ENDOR studies have confirmed the identification (Hage et al., 1986). In terms of the simple molecular orbital model used for silicon (Fig. 3-2), the center is observed in the neutral state which has three electrons in the t, orbital. It is therefore isoelectronic with Vii. Contrary to V g , however, the total spin is S = 3/2 and no Jahn-Teller distortion occurs. Here again, as for V, in diamond, the many-electron effects dominate in this wider bandgap semiconductor, and the orbitally nondegenerate 'A, state results. Annealing studies reveal that V,, in GaP disappears in a 15 min isochronal anneal at -350 " C . Detailed kinetic studies have not been performed. However, using again the simple arguments of Sec. 3.4.1.3 (Eq. (3-6)),we can make a rough estimate for its migration barrier of 1.5 eV. A level at E, + 0.64 eV has been tentatively identified with the single donor level E (O/+) of V,, by correlation of its production and anneal-

-

ing with the EPR results (Mooney and Kennedy, 1984). Theoretical calculations have been performed for V,, in GaAs which come to a surprising conclusion (Baraff and Schluter, 1986; Bar-Yam and Joannopoulos, 1986; Chadi, 1997). They predict that the gallium vacancy should exhibit instability as a neighbor As atom moves into the gallium site producing an AS,, antisite with an adjacent arsenic vacancy. Associated with this conversion is a strong negative-U property, the defect converting from a V,, multiple acceptor in n-type material to a (AS,, + VAS) multiple donor in p-type material. Chadi ( 1997) calculated similar behavior for V,, in GaN, and speculated that this is perhaps a common phenomenon in all III-V hosts. In Gap, observation by EPR and ENDOR of the neutral 4A2,S = 3/2, V,, state, which according to Chadi should be metastable, argues against such behavior i n that material. However, it is interesting to speculate as to whether the PP3X or PP3Y P,,-related centers discussed in Sec. 3.5.1.1 could possibly be the analogous PGa+V , configuration for the gallium vacancy (Beall et al., 1984). Clearly further discussion concerning the metal vacancies in IIIV materials is premature at this juncture.

3.5.3 Metal Interstitials An S = 112 ODEPR spectrum has been reported in as-grown liquid encapsulated (LEC) GaP that has been attributed to an isolated gallium interstitial (Gay) (Lee, 1988). Strong isotropic hyperfine interactions with the two naturally abundant 69Ga and 'lGa isotopes are partially resolved as expected for a tightly bound S state (a,) of Ga;' on-center i n the tetrahedral interstitial site. Two triplet (S = 1) low symmetry centers have also been reported in ODEPR studies


of luminescence in GaP that have been assigned to Gai-related centers. One, labeled here as Ga,X was originally attributed to an oxygen impurity (Gal et al., 1979), but has more recently been assigned to a Gai-defect complex (Lee, 1988; Godlewski and Monemar, 1988). Another, labeled GaiY,has been observed in Cu- and Li-doped GaP (Chen and Monemar, 1989). Both reveal the characteristic hyperfine of the gallium nucleus expected for an excited S = 1 center involving the gallium interstitial. In molecular beam (MBE) and organometallic vapor phase (OMVPE) epitaxially grown Al,Ga,-,As, an S = 1/2 ODEPR spectrum detected in luminescence has also been reported that has been attributed to Ga:+ (Kennedy et al., 1988). Here the partially resolved gallium hyperfine interaction displays anisotropy which could result simply from the alloy disorder or could also be evidence for a nearby defect. The spectral dependence of the luminescence associated with the center has led to a tentative assignment of its second donor state E(+/++),if isolated, at -Ev+ 0.5 eV. A very similar, but isotropic, center has also been observed in luminescence by ODEPR in GaAs/AIAs superlattices (Trombetta et al., 1991) and in GaAs/AIGaAs heterostructures (Wimbauer et al., 1997b). In the latter case, the authors also detected the EPR signal electrically. Finally, ODEPR studies in electron-irradiated wurtzite GaN have detected two centers which display anisotropic gallium hyperfine interactions, which have also been tentatively identified with Gq-related defects (Linde et al., 1997; Bozdog et al., 1999). In each of the above cases, identification as involving Ga:+ is possibly correct but it must be remembered, as in the case of VIII antisites (Sec. 3.5.1.1) that an S-state is very insensitive to a nearby defect, and it is like-

153

ly therefore that the interstitial is involved in a complex with a nearby impurity or defect.

3.5.4 Defects on the Group-V Sublattice No direct magnetic resonance identification of any group-V vacancy or interstitial exists for a III-V semiconductor. Theory has resulted in some interesting suggestions, however, for the group-V vacancy. The results of Chadi (1997) predict negative-U behavior between on-center VAS in GaAs or on-center V, in GaN, each of which is a donor, and a distorted configuration in which the neighbors dimerize, producing an acceptor. Chadi speculates that this behavior (which, as already pointed out in Sec. 3.4.3, he also predicts for the anion vacancies in the II-VIS) might be common in all III-Vs. On the other hand, these defects are definitely produced by electron irradiation. In GaAs (Pons and Bourgoin, 1981) and InP (Massarani and Bourgoin, 1986), for example, the production of defects by displacements on the group-V atom sublattice have been established by electrical measurements. This has been accomplished by DLTS studies using bombarding electron energies very near the displacement threshold where the unique anisotropy of defect production expected for the group-V atom sublattice is observed. In GaAs, arguments have been presented that most of the several DLTS peaks are associated with the primary VAS+ Asi Frenkel pairs of different lattice separations that are frozen into the lattice and stable up to -500 K where they anneal primarily by recombining and annihilating themselves (Pons and Bourgoin, 1985; Bourgoin et al., 1988). Evidence of charge state effects on the annealing rates and recombinationenhanced processes for the pairs have been

154


presented (Stievenard and Bourgoin, 1986; Stievenard et al., 1986). A broad almost structureless EPR signal has been observed that anneals in the same temperature region as these DLTS levels which could possibly be associated therefore with these pairs. However, in the absence of resolved hyperfine interactions, this cannot be considered an identification. In the modeling of the annealing processes in GaAs, it has been suggested that it is the As, that is probably the mobile species. A spectroscopic piece of evidence that may be consistent with this is the observation in infrared absorption studies that room temperature electron irradiation produces new LVM spectra in both GaAs and GaP that can be identified as substitutional BGa and Cv impurities which have trapped a nearby defect (Newman and Woodhead, 1984). The vibrational frequencies suggest that the defect is interstitial. Identification with the group-V interstitial requires its separation from the Frenkel pair and migration at room temperature. It has been suggested that this could occur by recombination-enhanced processes accompanying the ionization associated with the long irradiation required for these studies or perhaps by an enhanced mobility in p-type material. Again, in the absence of specific benchmark identifications by magnetic resonance techniques, this scenario for defects on the group-V sublattice remains speculative.

3.6 Summary and Overview It is a difficult task to present a review of a rapidly evolving field such as this for a series intended to serve as a useful reference for years to come. It is this sobering fact that has dictated the decision here to concentrate primarily on what is believed to be well established. In doing this, we have left out

the results of many important experimental studies (DLTS, EPR, ODEPR, optical and infrared photo-luminescence and absorption, positron annihilation, etc.), which in many cases may well monitor the pristine intrinsic defects. The problem is that unless detailed microscopic structural information is available, for example, from resolved hyperfine structure, or isotope shifts of the immediate neighbors, it is not possible to sure when this is or is not the case. My apologies to the many scientists whose work has not been included due to this decision. From the many other studies, exciting and interesting models have been postulated for the intrinsic defects and their reaction processes. This is particularly true for the 111-V semiconductors where there has been a great deal of research activity but where magnetic resonance clarification has been somewhat less successful so far. There are two comprehensive reviews that deal with this subject for GaAs, the most thoroughly studied system. These are recommended to the interested reader who would like to explore further this fascinating subject (Pons and Bourgoin, 1985; Bourgoin et al., 1988). Still, even though we have limited ourselves only to the better established defects, it is clear that we have begun to accumulate valuable insight into the properties of the intrinsic defects in semiconductors. Vacancies, interstitials, and/or antisites have been identified and studied in a few representative cases throughout the semiconductor series from the covalent group-IV semiconductors, to the substantially ionic 11-VI’S. In this final section, we will first summarize briefly these cases, as presented in this chapter. We will then attempt an overview by pointing out a useful systematic pattern that they appear to fall into. This hopefully may serve as a guide to understanding, or predicting, the properties of all of the

3.6 Summary and Overview

intrinsic defects - including those not yet identified - throughout the semiconductor series.

3.6.1 Summary 3.6.1.1 Group IV Semiconductors In silicon, the isolated vacancy can take on five charge states in the forbidden gap (V=, V-, Vo, V+, V"). A simple one-electron molecular orbital model for the occupancy of the a, and t2 orbitals for the undistorted vacancy gives a good description of the electronic structure of the defect, the Jahn-Teller lattice relaxations involved, and the electrical level position for each of its charge states. The activation energy for migration is small, ranging from 0.18 to 0.45 eV depending upon charge state, and it migrates athermally under electronic excitation associated with e-h pair recombination at the defect. The vacancy has negative-U properties, the first donor level E (O/+) at E, + 0.03 eV, the second E (+I++) at E, + 0.13 eV. This plus the recombination-enhanced migrational properties follow as a natural consequence of the large Jahn-Teller distortions. The simple molecular orbital model also appears to work well for the vacancy in the other group-IV semiconductors, with the exception that many-electron interactions, which favor Hund's rule of occupancy of the levels, become more important as the bandgap increases. Interstitial silicon has not been observed directly by magnetic resonance techniques but a great deal is believed to be known about it from study of its trapped configurations and from theory. It is highly mobile (athermal) even at 4.2 K under electron irradiation conditions in p-type material. In n-type material its activation energy for migration must be 50.57 eV. Several near-

155

ly energetic interstitial configurations (in the interstices and in bonding arrangements) appear to exist for the interstitial. Theory predicts a conversion between them vs charge state and negative-U properties between Si?, which is stable in the tetrahedral site, and Si:, which prefers a bonded configuration. Alternation between the two sites accompanying e- and h+ capture provides an explanation for its athermal migration during electron irradiation. Little is known concerning interstitials in the other group-IV semiconductors, with the exception that indirect evidence also exists for their high mobility in germanium and diamond.

3.6.1.2 11-VI Semiconductors The metal vacancy produces a double acceptor level E (=/-) in most 11-VI materials (ZnS, ZnSe, CdS, CdTe, B e 0 and ZnO) and, in its single negatively charged state, undergoes a large static trigonal distortion. In ZnS and ZnSe, complete CC diagrams have been constructed that describe the electrical and optical properties of the defects and identify the distortion as JahnTeller in origin. For ZnS and ZnSe, the second acceptor levels E (=/-) are at E,+ 1.1 eV and Ev + 0.66 eV, and the migration energy barriers are 1.04 eV and 1.25 eV, respectively. The chalcogen vacancy is a deep double donor, its singly ionized V + state being S-like, spread equally over its four group-I1 neighbors. It is stable at room temperature. Theory predicts that both vacancies may be amphoteric, the metal vacancy converting to a double donor and the chalcogen vacancy converting to a double acceptor, as a result of large lattice distortions accompanying the charge state changes. No direct experimental evidence of either exists at present.

156


Interstitial zinc in ZnSe can be found oncenter in the tetrahedral interstitial site surrounded by either four Se atoms, (Zn&,, or four zinc atoms, (ZniJz,. It is a double donor with its second donor state E(+/++)deep at -Ec- 0.9 eV for (Zn,),, and at -E,- 1.5 eV for (Zni)zn.Its thermally activated migration energy is -0.6-0.7 eV, but it can migrate athermally by cycling between the two interstitial sites under electronic excitation.

3.6.1.3 Ill-V Semiconductors The VI,, antisite is a double donor, which is present in as-grown materials. It has been observed by magnetic resonance techniques in Gap, GaAs, and InP in its singly positively charged state. In this charge state, the group-V atom is on-center in the group-I11 atom site with the unpaired electron highly localized in an S-like orbital centered on the atom. Interstitial Gar* also appears to have been detected in as-grown Gap, Al,Ga,-,As, GaAs/AlGaAs heterostructures, and electron-irradiated GaN. For it the unpaired electron is similarly in a deep S-orbital centered on the atom. For both the VI,, antistite and interstitial Ga:+ the isotropic character of their S-electronic states makes it difficult to distinguish whether the defects being studied are truly isolated or paired with another impurity or defect. Although this distinction is important in identifying the defect, the chemistry of its formation, and its electrical properties. the insensitivity to complexing assures us on the other hand that the electronic sfrucrure of the isolated defect is being correctly deduced. The neutral group-111 atom vacancy VI,, has been observed in electron-irradiated GaP in the high spin, S = 3/2 state. In this state the defect displays the full tetrahedral

(Td) symmetry of the undistorted lattice. The vacancy is stable only to 350 "C. As for the 11-VIS, theory again predicts that the vacancies on either sublattice may be amphoteric, converting from donor to acceptor for the group-V atom vacancy, or vice versa for the group-I11 vacancy, as a result of large lattice rearrangements accompanying the corresponding charge state changes. Similar structural instabilities are also predicted for both antisites. The predicted instability for the AsGa antisite in GaAs appears to have been experimentally confirmed, giving rise to the much studied metastable EL2 defect. No experimental evidence of the other predicted instabilities exists at present, however.

3.6.2 Overview In the case of the silicon vacancy and the metal vacancy in 11-VI materials, it was found instructive to construct a singly degenerate (a,) and a triply degenerate (t2) molecular orbital for the defect in its undistorted T, lattice arrangement which could be populated by the appropriate number of electrons for each defect charge state. This served to account for the cases where large symmetry lowering lattice relaxations are encountered as being the result of electronic degeneracy and a consequent Jahn-Teller distortion. These distortions in turn were found to have important consequences for the electrical level positions, the optical transitions, and the migrational properties of the defect. Let us attempt here therefore to extend this approach to all of the intrinsic defects throughout the semiconductor series to see what insight this might provide. For a vacancy on either sublattice, one can always construct a symmetric combination of the ruptured bonds on the nearest neighbors of the other sublattice to form an a, orbital, and

157


three equally energetic combinations (antisymmetric by pairs) to form the t2 orbitals, higher in energy, as was described first for the silicon vacancy in Section 3.2.2.1. This is reproduced in Fig. 3-14a. This can also be done for any interstitial as well, where the al state now represents the outer s-orbital, and the t2 states, the outer p-orbitals, of the interstitial atom. In this simple approach, therefore, the only qualitative difference between a vacancy on either sublattice or an interstitial in any semiconductor is the electron occupancy and the positions of the occupied and unoccupied al and t2 orbitals with respect to the band gap.

3.6.2.1 Vacancies In Table 3-3 we summarize the al and t2 level occupancies for the various charge states of vacancies in the group IV, 111-V, and 11-VI semiconductors. We have included also the ionic I-VI1 alkali halides for the further insight that they can provide. The configuration in each case is denoted ayt;, where m is the number of electrons in the a, orbital and n is the number in the t2 orbital. We have assumed here that the strong crystal-field limit applies (the a, - t2 splitting is greater than the electron-electron interactions) and we fill the a, level before occupying the t2 levels. Orbital degeneracy occurs only for the vacancies which have partially filled t2 shells, i.e., n = 1 to 5 . For the others, Jahn-Teller symmetry lowering distortions should not occur and we anticipate full Td symmetry for the defect. This has been confirmed directly for V c I in the 11-VI mate[the F-center (Klick, 1972)] rials and in the I-VI1 alkali halides, and inferred indirectly for V i in 11-VI'S (Fig. 3-10) and V y t in silicon (Fig. 3-2). Accordingly, we can expect full Td symmetry also for vi:, V t , and V,"i+,++ in the corresponding semiconductors.

VtlI

/ /

(bl

(a)

,

,-

a1

/

(C)

Figure 3-14. One-electron molecular orbitals in undistorted (Td) symmetry for a) vacancy or host interstitial, b) VI,, antisite, c) 111, antisite.

Table 3-3. Elecron occupancies (art;) for the various charge states of vacancies in the group IV, 111-V, 11-VI, and I-VI1 materials. rn

n

I

I1

111

IV

V

VI

VI1

JahnTeller

2 2

1 0

1 0

0 0

+ ++

Experimentally observed occupancies are shown in bold print.

I

0

+

0

+ ++

0

+

158


On the other hand, we must be prepared for symmetry lowering Jahn-Teller distortions for the other cases shown in the table where partial occupancy of the t2 levels occurs. This has already been confirmed for the V i metal vacancies i n 11-VI’S and the various charge states of the vacancy in silicon. The alkali halide extreme also confirms this for Vp [V, center (Kabler, 1972)]. Consider, therefore the group-111vacancy in a 111-V semiconductor, about which we have no direct experimental information. Table 111predicts that Vi; is stable in full T, symmetry, but that ionization to VTII and to the less negatively charged states should produce the tendency for Jahn-Teller distortions which increase progressively vs. ionization as it goes from aft: to aft:, etc. This provides a clear physical mechanism for the theoretical predictions for the structural instability of the 111-V cation vacancy described in Sec. 3.5.2. An inward trigonal distortion of one of the group-V atom neighbors can be expected for VGI just as has been experimentally established for V i (both aft:). The distortion should then increase with further ionization to V i l (a:ti), Vyll (a:t:), etc., being driven by the increased Jahn-Teller coupling to two, then three holes, etc. In this way, the group-V atom is propelled inward toward the vacant group-I11 atom site, leaving an adjacent arsenic vacancy behind, converting the defect to a positive donor, as predicted. The similar theoretical predictions described in Sec. 3.4.3 for the metal vacancies in 11-VIS are also easily understood as they go from aft: (v;)to aft: (v:), etc. Contrary to this simple picture, however, the neutral gallium vacancy in Gap, the negative vacancy in diamond, and the negative silicon vacancy in S i c have all been observed i n the closed shell 4A2, high spin S = 312 state with full T, symmetry even though their configurations are a:ti, the same as

for the strongly distorted S = 1/2 negative vacancy in silicon. This means that for these particular larger bandgap materials, for which the wave functions can be more highly localized, the many-electron effects which favor Hund’s rule occupancy are beginning to compete with the energy gain from the Jahn-Teller distortions. It is for this half-filled t i configuration that the competition will tend to show up first, and where it will have the strongest consequences, because the symmetry lowering distortions vanish. The many-electron interactions can continue to be important for the t i and t; configurations, but they cannot remove the degeneracy, and Jahn-Teller distortions will still tend to occur. Table 3-3 provides a reliable, simple, physical, and predictive explanation of the electronic and lattice structures for the zero particle (t:, t:) and for the single particle (ti, $1 configurations of vacancies in all semiconductors, and for the other configurations (t;, t;, ti) in silicon and (most likely) the other modest bandgap materials. As the bandgap increases, however, manyelectron effects must be increasingly considered in the occupancy of the levels. For the t; and ti configurations, the prediction of Jahn-Teller distortions should remain valid, but the specific distortions that occur and their strengths may be altered. An example here is the unusual (100) dihedral distortion for the S = 1 a: ti state of the neutral group-I1 vacancy in B e 0 and ZnO, as discussed in Sec. 3.4.2.1. For the ti configuration, the many-electron effects, if large enough, can completely quench the JahnTeller distortions. Unfortunately, the modern local density calculations do not take into account all of the relevant parts of the electron-electron interactions that are necessary to properly address the many-electron occupancy problem and so far their results appear to follow


the simple one-electron predictions of Table 3-3. As the bandgap of the semiconductor in question increases, however, it is clear that their predictions can be expected to begin to fail, unless this limitation is corrected. This represents a formidable challenge to theory, but a necessary one. One way or the other, when close competition exists between the electron-electron interactions and the Jahn-Teller instabilities, we can expect interesting new effects involving metastability arising from the two or more closely energetic but differently coupled spin states which have different lattice relaxations, and theory will be essential in guiding us.

3.6.2.2 Interstitials In Table 3-4 we present the corresponding al (s) and t2 (p) occupancies predicted for host interstitials. Again as predicted, the two observed cases, Zn; and G C , with a; t: configurations, have the full Td symmetry. The only other experimentally observed case is the interstitial halogen atom in alkali halides [H-center (Kabler, 1972)] and, as predicted for its aft; configuration, it is strongly distorted. [Note added in proof, 12/99: The (100)-split configuration recently reported for neutral interstitial carbon in

159

diamond (Hunt et al., 1999) also follows from its a:t; configuration.] This molecular orbital approach also provides a logical explanation for the theoretically predicted properties for the silicon interstitial. The predicted on-center tetrahedral site for Si:+ follows naturally from its aft: configuration. Its movement off this site in the Si: and Si; states in turn follows naturally from the partial t, occupancy, the negative-U properties reflecting the much stronger Jahn-Teller driving force (x2) for the distortion in Sip with a2tS. From Table 3-4, we are led to anticipate similar interesting effects for the Group-V interstitials in 111-V materials, about which we have no experimental information, and also the Group-VI interstitials in 11-VI'S. In this latter case, the off-center bonding configuration inferred for the tellurium interstitial in ZnTe, and described in Sec. 3.4.2.3, is consistent with this. The many-electron effects may not be so important for interstitials because of the large electron-lattice (Jahn-Teller) couplings associated with the strong bonding energies available to valence p-orbitals. Therefore, the simple one-electron occupancy models may turn out to be reliable over a larger range of materials.

Table 3-4.Electron occupancies (s"p") for the charge states of host interstitial atoms in the group IV, 111-V, 11-VI, and I-VI1 materials.

0

0 -

0

0

0

0

+ ~

~~

+

+ ++

+ ++

+f

~

Experimentally observed occupancies are shown in bold print.

0

+ ++

+

++

JahnTeller

160


3.6.2.3 Antisites The antisites can also be cast into this molecular orbital picture, with the aid of Fig. 3-14b and c. Here the antisites are being viewed as simple substitutional double donors (or acceptors) and treated therefore as similar to host atoms but with an increase (or decrease) in nuclear charge of 2e. As normal host atoms, theirs and p orbitals interact, respectively, with the a, and t2 vacancy orbitals of the neighbors forming bonding states pushed downward into the valence band and antibonding states shifted upward into the conduction band, leaving the gap empty (i.e., “healing” the vacancy). However, increasing the nuclear charge as in the VI,, antisite serves to pull the antibonding states down, the a , state returning into the gap as depicted in Fig. 3-14b. Similarly, decreasing the nuclear charge, as for the IIIv antisite, permits the bonding t2 states to rise back into the gap as shown in Fig. 3-14c. Referring to Fig. 3-14c, the theoretically predicted instability for the 111, antisite again follows naturally. For the double minus state, the t, level is filled and the group-I11 atom resides on-center in the group-V atom site. The singly ionized state 111; is tl and Jahn-Teller distorted, the group-I11 atom displacing trigonally toward the interstitial site. In the neutral state 1110, (t;) the Jahn-Teller coupling is doubled, the atom displacing farther toward the interstitial position. Again as for the silicon vacancy, Eqs. (3-1) to (3-3) in Sec. 3.2.2.3, negative-U can result between the neutral and doubly negative states. Referring to Fig. 3-14b, the VI,, antisite should remain undistorted for its three donor charge states: V i I (a:), ViIl (at), and V;: (a:). This again has been confirmed directly for the paramagnetic qI1 state studied by EPR, ODEPR, and ODENDOR. The

figure also provides an explanation for the theoretically predicted metastable EL2 behavior for the isolated VI,, antisite, as described in Sec. 3.5.1.1. Occupying the t2 level in the conduction band provides an excited state for the neutral defect (aft: or ayt;) which can Jahn-Teller distort in a similar fashion to that for 111, above, the now central V atom injecting into the interstitial position as shown in Fig. 3-13b. In this case, the Jahn-Teller relaxation energy is not quite sufficient to overcome the al- t2 promotion energy and a long-lived metastable neutral state is formed.

3.6.2.4 Migration Barriers Migration barrier energies for the vacancies and interstitials in silicon have been found to be small (-0-0.6 eV) and charge state dependent. Under recombination conditions, migration for both is athermal. For the compound semiconductors, we expect somewhat higher barriers, motion on one sublattice being hindered by the presence of atoms on the other. This is borne out for metal vacancies in ZnS (1.04 eV), ZnSe (1.26 eV) and GaP (-1.5 eV) and also for the zinc interstitial in ZnSe (-0.6-0.7 eV). The latter can also be made to migrate athermally under recombination conditions. In all cases where large lattice relaxational changes are predicted vs charge state, we may anticipate strong charge state dependence and recombination-enhancement contributions to the diffusion mechanisms for the defects.

3.7 Acknowledgements The original review (Watkins, 1991) was made possible by support from National Science Foundation grant DMR-89-02572 and Office of Naval Research Electronics

3.8 References

and Solid State Program grant NOOOO14-90J-1264. This updated version was made possible by support from National Science Foundation grant DMR-97-04386 and Office of Naval Research Electronics and Solid State Program grant N000014-94-1-0117.

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Trombetta, J. M., Kennedy, T. A., Tseng, W., Gammon, D. (1991). Phys. Rev. B43,2458. Troxell, J. R. (l979), PhD Dissertation, Lehigh University, unpublished. Troxell, J. R., Watkins, G. D. (1980), Phys. Rev. B 22, 921. Troxell, J. R., Chatterjee, A. P., Watkins, G. D., kmerling, L. C. (1979), Phys. Rev. B 19, 5336. Twitchen, D. J., Newton, M. E., Baker, J. M., Tucker, 0. D., Anthony, T. T., Banholzer, W. F. (1996), Phys. Rev. B 54,6988. Twitchen, D. J., Newton, M. E., Baker, J. M., Anthony, T. R., Banholzer, W. F.(1999), Phys. Rev. B 59, 12900. Uftring, S. J. (1998), Ph.D. Dissertation, Lehigh University, unpublished. Van de Walle, C. G., Blochl, P. E. (1993), Phys. Rev. B 47,4244. Van de Walle, C. G., Neugebauer, J. ( 1995), Phys. Rev. B 52, R14320. \an Oort, E., Manson, N. B., Glasbeek, M. (1988), J. Phys. C: Solid State Phys. 21, 4385. van Wyk, J. A., Tucker, G. D., Newton, M. E., Baker, J. M., Woods, G. S., Spear, P. (1995), Phys. Rev. B 52, 12657. Von Bardeleben, H. J., Stievenard, D., Deresmes, D., Huber, A., Bourgoin, J. C. (1986), Phys. Rev. B 34, 7192. Watkins, G. D., Corbett, J. W. (1961), Phys. Rev. 121, 1001. Watkins, G . D. ( 1964), in: Radiation Damage in Semiconductors, Paris: Dunod, 97. Watkins, G . D. (1967), Phys. Rev. 155, 802. Watkins, G. D. (1969). IEEE Trans. Nucl. Sci. NS-16, 13. Watkins, G . D. (1971), in: Radiation Effects in Semiconductors: Corbett, J. W., Watkins, G. D. (Eds.). New York: Gordon and Breach, 301. Watkins, G . D. (1974). Phys., Rev. Lett. 33, 223. Watkins. G . D. (1975a), in: Point Defects in Solids, Vol 2: Semiconductors and Molecular Crystals: Crawford, J. H. Jr., Slifkin, L. M. (Eds.) New York: Plenum, pp. 333-392. Watluns, G. D. (1975b), in: Lattice Defects in Semiconductors 1974: Huntley, F. A. (Ed.). London: Inst. Phys. Conf. Se. No. 23, 1. Watkins, G. D. (1975c), Phys. Rev. B 12, 4383. Watkins, G. D. (1975d), Phys. Rev. B 12, 5824. Watkins, G . D. (1975e), in: Lattice Defects in Semiconductors 1974: Huntley, F. A. (Ed.). London: Inst. Phys., Conf. Se. No. 23, 338. Watkins, G. D. (19750, Intrinsic Defects in II-VI Compounds (ARL TR 75-001 1). Clearinghouse, Springfield, VA 2215 1 USA: National Technical Information Services. Watkins, G . D. (1976a), in: Point Defects in Solids; V01.2 Semiconductors and Molecular Crystals: Crawford, J. H. Jr. and Slifkins, L. M. (Eds.). New York: Plenum, Ch. 4.

3.8 References

Watkins, G. D. (1976b), Phys. Rev. B 13, 251 1. Watkins, G. D. (1977), in: Radiation Effects in Semiconductors 1976. Urli, N. B., Corbett, J. W. (Eds.). London: Inst. Phys., Conf. Se. No. 31,95. Watkins, G. D. (1983), Physica 117B-l18B, 9. Watkins, G. D. (1984), in: Festkorperprobleme XXN: Grosse, P. (Ed.). Braunschweig: Vieweg, pp. 163-1 89. Watkins, G. D. (1986), in: Deep Centers in Semiconductors, Pantelides, S . T. (Ed.). New York: Gordon and Breach, pp. 147-183. Watkins, G. D. (1989), Rad. Eff:and Defects in Sol. 111-112,487. Watkins, G. D. (1990), in: Defect Control in Semiconductors: Sumino, K. (Ed.). Amsterdam: North Holland, pp. 933-941. Watkins, G. D. (1991), in Electronic Structure and Properties of Semiconductors: Schroter, W. (Ed.), Weinheim: VCH, Chap. 3. Watkins, G. D. (1997), in Defects and Diffusion in Silicon Processing: de la Rubia, T. D., Coffa, S., Stolk, P. A., Rafferty, C. S. (Eds.). Pittsburgh: Mat. Res. SOC.Vol. 469, p. 139. Watkins, G, D, (1998), in Identification of Defects in Semiconductors: Stavola, M. (Ed.), Semiconductors and Semimetals, Vol. 51A. San Diego: Academic, Chap. I . Watkins, G. D. (1999), in EMIS Data Review No. 20: Properties of Crystalline Silicon: Hull, R. (Ed.). London: IEE, pp. 643-652. Watkins, G. D., Corbett, J. W. (1964), Phys. Rev. 134, A 1359. Watkins, G. D., Corbett, J. W. (1965), Phys. Rev. 138, A 543. Watkins, G. D., Brower, K. L. (1976), Phys. Rev. Lett. 36, 1329. Watkins, G. D., Troxell, J. R. (1980), Phys. Rev. Lett. 44, 593. Watkins, G. D., Troxell, J. R., Chatterjee, A. P. (1979), in: Defects and Radiation Effects in Semiconductors 1978: Albany, J. H. (Ed.). London: Inst. Phys., Conf. Se. No. 46, p. 16. Watkins, G. D., Chatterjee, A. P., Harris, R. D., Troxell, J. R. (1983), Semicond. Insul. 5, 321. Watkins, G. D., Rong, F., Barry, W. A., Donegan, J. F. (1988), in: Defects in Electronic Materials: Stavola, M., Pearton, S. J., Davies, G. (Eds.). Pittsburgh: Mat. Res. SOC.,Vol. 104, 3. Weber, E. R., Omling, P. (1985), in: Festkorperprobleme X X V ; Grosse, P. (Ed.). Braunschweig: Vieweg, p. 623. Whan, R. E., (1965), Appl. Phys. Lett. 6, 221. Wimbauer, T., Meyer, B. K., Hofstaetter, A., Scharmann, A., Overhoff, H. (1997a), Phys. Rev. B 56, 7384.

165

Wimbauer, T., Brandt, M. S., Bayerl, M. W., Stutzmann, M., Hofmann, D. M., Mochizuki, Y., Mizuta, M. (1997b), Matel: Sci. Forum 258-263, 1309. Wirbeleit, F., Niklas, J. R. (1997), Mater: Sci. Forum 258-263,987. Woodhead, J., Newman, R. C., Grant, I., Rumsby, D., Ware, R. M. (1983), J . Phys. C 16, 5523. Yu, P. W. (1983), Phys. Rev. B 27, 7779. Zhan, X. D., Watkins, G. D. (1993), Phys. Rev. B 47, 6363. Zhang, S. B., Chadi, D. J. (1990/, Phys. Rev. Lett. 64, 1789. Zheng, J. F., Stavola, M., Watkins, G. D. (1995), in: 22nd Int. Con$ on the Physics of Semiconductors, Lockwood, D. J. (Ed.). Singapore: World Scientific, pp. 2363-2366.

General Reading Bourgoin, J. C., von Bardeleben, H. J., Stievenard, D. (1988), “Native defects in gallium arsenide”, J. Appl. Phys. 64, R65-R91. Pons, D., Bourgoin, J. C. (1985), “Irradiation-induced defects in GaAs”, J. Phys. C: Solid State Phys. 18, 3839-387 1. Spaeth, J.-M., Niklas, J. R., Bartram, R. H. (1992), Structural Analysis of Point Defects in Solids. Berlin: Springer-Verlag. Watkins, G. D. (1976) “EPR Studies of Lattice Defects in Semiconductors”, in: Defects and their Structures in Nonmetallic Solids: Henderson, B., Hughes, A. E. (Eds.). New York: Plenum, pp. 203220. Watkins, G. D. (1984), “Negative-U Properties for Defects in Solids”, in: Festkorperprobleme X X N Grosse, P. (Ed.). Braunschweig: Vieweg, pp. 163189. Watkins, G. D. (1986), “The Lattice Vacancy in Silicon”, in: Deep Centers in Semiconductors: Pantelides, S . T. (Ed.). New York: Gordon and Breach, pp. 147-183. Watkins, G. D. (1990), “Intrinsic Defects on the Metal Sublattice in ZnSe”, in: Defect Control in Semiconductors: Sumino, K. (Ed.). Amsterdam: North Holland, Vol. 1, pp. 933-941. Watkins, G. D. (1998), “EPR and ENDOR Studies of Defects in Semiconductors”, in: Identification of Defects in Semiconductors: Stavola, M. (Ed.), Vol. 5 1A of Semiconductors and Semimetals. San Diego: Academic Press, Chap. 1.

4 Deep Centers in Semiconductors Helmut Feichtinger Institut fur Experimentalphysik der Karl.Franzens.Universitat. Graz. Austria

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.1.1 Shallow and Deep Impurities: Technological and Physical Relevance . . . 170 4.1.2 The Identification Problem and the Localization-Delocalization Puzzle . . 172 Deep Centers: Electronic Transitions and Concepts . . . . . . . . . . . 174 4.2 4.2.1 Ionization at Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . 174 4.2.2 Franck-Condon Transitions and Relaxation . . . . . . . . . . . . . . . . . 176 4.3 Phenomenological Models and Electronic Structure . . . . . . . . . . . 180 4.3.1 The Point-Ion Crystal Field Model . . . . . . . . . . . . . . . . . . . . . . 180 4.3.2 The Defect Molecule Picture . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.3.2.1 Example: Nitrogen in Gallium Phosphide . . . . . . . . . . . . . . . . . . 183 185 4.3.2.2 Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Transition Metals: Results of Quantitative Calculations . . . . . . . . . . . 187 4.3.3.1 Gap Levels and High Spin-Low Spin Ordering . . . . . . . . . . . . . . . 187 4.3.3.2 Coulomb Induced Nonlinear Screening and Self-Regulating Response . . . 192 4.3.4 Ionization Energies and Trends . . . . . . . . . . . . . . . . . . . . . . . 194 194 4.3.4.1 Transition Metals in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4.2 Compound Semiconductors and Bulk References . . . . . . . . . . . . . . 197 199 4.3.5 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 4.3.5.1 Internal Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.3.5.2 Rydberg-Like States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Properties of Selected Systems . . . . . . . . . . . . . . . . . . . . . . . 204 4.4.1 Chalcogens in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 4.4.1.1 Sulfur, Selenium, and Tellurium in Silicon . . . . . . . . . . . . . . . . . 204 206 4.4.1.2 Oxygen and Nitrogen in Silicon . . . . . . . . . . . . . . . . . . . . . . . DX Centers in AI,Ga,, As . . . . . . . . . . . . . . . . . . . . . . . . . . 207 4.4.2 4.4.2.1 Large Lattice Relaxation and Metastability . . . . . . . . . . . . . . . . . 207 4.4.2.2 Microscopic Models for DX Centers . . . . . . . . . . . . . . . . . . . . . 210 4.4.3 Deep Transition Metal Donor-Shallow Acceptor Pairs in Silicon . . . . . 213 213 4.4.3.1 Electronic Structure and Trends . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.2 Charge State Controlled Metastability . . . . . . . . . . . . . . . . . . . . 214 4.4.4 Thermal Donors in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.4.5 Hydrogen Passivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Appendix: Ionization Energies and Level Positions 4.5 of Isolated Transition Metal Impurities in Silicon . . . . . . . . . . . . 222 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.6

168

4 Deep Centers in Semiconductors

List of Symbols and Abbreviations hyperfine parameter magnetic field neutral and ionized donors energy of absorption and emission pair binding energy barrier height for capture rate energy of the conduction and valence band edges binding energy of shallow donors Fermi energy migration energy phonon energy of state n optical and thermal ionization energy occupancy level lattice relaxation energy zero phonon transition energy transition energy electron Gibbs free enthalpy standard chemical potential for ionization spectroscopic splitting factor degeneracy factor enthalpy ionization enthalpy Planck constant reduced Planck constant spin of nucleus total angular momentum Boltzmann’s constant wave vector orbital angular momentum orbital and nuclear quantum number acceptor and donor concentration effective density of states pressure hole concentration configurational coordinate charge state of the defect Huang-Rhys factor entropy total electron spin ionization entropy temperature Mott-Hubbard correlation energy


V V(r> V

(v>n,p X

z

crystal volume defect potential anion-cation transfer matrix element mean thermal velocity mole fraction valency number

w

ratio of the radiative recombination rate to the total recombination rate chemical potential of an electron frequency thermal-capture cross section for electrons (n) or holes (p) optical cross section radiative or non-radiative lifetimes photon flux angular frequency

DLTS DRAM EMT ENDOR EPR EXAFS FIR FTIR LCAO LDOS LED LLR LMTO- ASA MOD- FET MOS NAA ODMR OSB PTIS TD VLSI Y LID ND STD

deep level transient spectroscopy dynamic random access memory effective mass theory electron nuclear double resonance electron paramagnetic resonance extended X-ray absorption fine structure spectroscopy far infrared spectroscopy Fourier transform infrared transmission spectroscopy linear combination of atomic orbitals local density of states light-emitting diode large lattice relaxation linearized-muffin-tin-orbital in atomic-spheres approximation modulation field effect transistor metal oxide semiconductor neutron activation analysis optically detected magnetic resonance Ourmazd-Schroter-Bourret model photo-thermal ionization spectroscopy thermal donors very large scale integration Y-shaped thermal donor core new donors shallow thermal donors

rl

A Y %p

%P rI.3

4J

Gr

169

170

4 DeeD Centers in Semiconductors

4.1 Introduction 4.1.1 Shallow and Deep Impurities: Technological and Physical Relevance Among defects in semiconductors, those impurities that are widely used as dopants to control type and resistivity are well known as “shallow” donors or acceptors. Their ionization energies are very small compared to the fundamental gap of a given semiconductor (Fig. 4-1). Hence, at room temperature and well below, shallow donors are depleted from their electrons, and acceptors, from their holes. Such centers are well understood in terms of effective mass theory (EMT) (Kohn, 1957, Pantelides, 1978). By adopting two basic assumptions, this reduces the boundstate equation for shallow defects to a hydrogenic Schrodinger equation with the potential V ( r ) being a screened Coulomb potential, and the free-electron mass replaced by an effective mass depending on the energy band structure of the host matrix (see Chapter 1 in this volume), where the isolated defect is embedded. In principle, the two assumptions require 1 ) the binding potential to be weak and to vary very slowly between two lattice points at a distance from the defect where static Conduction b a n d

“Shallow” donor

“Deep” level

*Ew

”Shallow” acceptor

dielectric screening is meaningful and 2) the particle wavefunction to be localized in k-space. Thus the bound states are derived exclusively from nearest-band-edge states. For a shallow substitutional impurity, EMT then gives a hydrogen-like Rydberg series, terminated at the energy of the conduction band edge (for donors) or the valence band edge (for acceptors). The binding energy for a donor has the form

ED =

2 4 * 1 Z e me (4nE E n ) ? 2 h2 n2

with a corresponding Bohr radius that extends over many lattice constants even for the 1 s ground state. According to the simplified picture of Eq. (4-1) for a fixed host, Rydberg series of different shallow centers should be of similar structure. Referring to the “deep” level of Fig. 4-1, it is easy to see that under steady-state conditions the electrical activity of the deep center depends on shallow doping and temperature. As an example, if the impurity visualized in Fig. 4-1 of concentration ND acts as a deep donor in p-type material, it will compensate the shallow acceptors to N,-ND = p , the free hole concentration at room temperature. If the same impurity is incorporated into n-type material, it would be of practically no influence unless the shallow background doping is almost negligible compared to the deep donor concentration. Otherwise, the impurity will tend to increase the free-electron concentration, depending more or less on temperature. Similar arguments hold if the deep impurity acts as an acceptor in the lower half of the band gap. Thus it becomes clear that the resistivitv. or even the tvDe of semiconductor in extreme cases, may be influenced by deep centers. However, if these effects are unwanted, they may be avoided by keeping deep im. I

Valence b a n d Figure 4-1. Simplified energy level scheme and ionization energies.

(4- 1 )

-

I

4.1 Introduction

purity contamination well below the shallow background doping. But even small amounts of deep impurities present in a semiconductor may greatly affect the carrier lifetime, which is an important parameter in all semiconductor devices. This is because non-radiative recombination proceeds most easily via deep centers since they can exchange carriers with both the conduction band and the valence band. But these effects of deep impurities such as gold and platinum are the ones exploited to reduce minority carrier lifetime in fast switching devices like thyristors (Lisiak and Milnes, 1975). Also, deep centers are used for extrinsic photoexcitation in photoconductors and infrared detectors (Sze, 1981; Sclar, 198 1). For light-emitting diodes (LED) the quantum efficiency in an indirect gap material like GaP is strongly enhanced by the substitutional impurity nitrogen which is isoelectronic to phosphorus. On the other hand the ratio q of the radiative recombination rate to the total recombination rate is given by (4-2) where the subscripts nr and r stand for nonradiative and radiative lifetimes. In indirect gap materials, it often appears that t,S tnr, with the major LED deficiency in the case of GaP coming from the non-radiative recombination path via two not totally identified deep centers (Peaker and Hamilton, 1986). In modern VLSI technology deep impurities may cause serious device problems, as they may act as generation or recombination centers in depleted or undepleted regions, respectively. Thus a series of device parameters in bipolar and MOS devices is degraded (Keenan and Larrabee, 1983). Apart from harmful effects due to the

171

electrical activity of deep centers, they are often also responsible (even in electrically inactive form) for limiting the diffusion length, increasing leakage, or reducing and softening reverse breakdown voltages when precipitated in active layers. It was shown, for example, that gate oxide degradation in silicon MOS DRAM processing results to a large extent from the interaction of grown-in heavy metal contamination (preferably fast-diffusing species like Cu, Pd, Ni) (Bergholz et al., 1989). Still, increasing large scale integration in device manufacturing makes it highly desirable to keep metals in the starting silicon material well below concentrations of 1 0 ' ~~ m - ~ . The physical nature of deep centers is reflected essentially, but by far not completely, by their deep ionization levels. Nevertheless, an initial approach may be to start again with shallow centers where the shortrange central cell potential is responsible only for a chemical shift, especially of ground state terms, and hence for the chemical trends in the rather small ionization energies. As the energy levels become deeper, this should be a consequence of an increased short-range potential which tends to confine the particle wave function around the defect site. Finally, one may arrive at a situation, where a particle can be bound, at least in its ground state, by the short-range potential alone, whereas the Coulomb part of the defect potential plays only an almost negligible role. In other words, contrary to shallow centers, the bound state of a deep center is a spatially compact state, hence delocalized in k-space, and therefore is not exclusively derived from the nearest-bandedge states (Hjalmarson et al., 1980). This statement is manifested in Fig. 4-2 showing how the binding energy (ionization energy, see Sec. 4.2.1) depends on central cell potentials for different impurity-host systems

172


(€&-€Ls,)in eV

Atomic electronegativity 0 -2 -I -6 -8

-10

Host

I

\

I I

\

\

I o

\ Te'

\

\

\

\

\

\

\

\

\

\

\ Se'

\

Figure 4-2. Experimental ionization energies of donors in Si and GaP :.( Cation site; 0 : Anion site) versus the central cell impurity potential strength, the latter given in differences between the s-orbital energy of the impurity atom (E;mp)and the replaced host atom (After Vogl, 1984).

(Vogl, 1984). Binding energies increase only slightly for rather small attractive central cell potentials (shallow centers) but steeply for large potentials (deep centers), the threshold separating the regions where either the long-range Coulomb tail or the short-range central cell potential binds an electron. 4.1.2 The Identification Problem and the Localization-Delocalization Puzzle Deep levels are induced by substitutional impurities (where a host atom is replaced) that do not belong to the groups in the periodic table lying closest to that (or those, in the case of compound semiconductors) of

the host crystal. But interstitial impurities and defect associates may also create deep levels. Here, transition metals play a key role in understanding deep impurity phenomena. Generally, this is also true for intrinsic defects like vacancies and selfinterstitials. Therefore, intrinsic defects, their electronic structure, and first-generation interactions with other impurities are treated separately i n the preceding chapter by G. D. Watkins. One may literally say that shallow centers are rather exceptional, as can be seen from compiled data (e.g. Sze, 198 1, p. 27). The vast number of impurity or defect systems gives rise to serious identification difficulties. These systems may differ in atomic structure, in host sites and bonding symmetries, in solid solubilities, in diffusivities, and in reactivities for interaction with other intrinsic or extrinsic defects present in the host crystal. This therefore results in very different sensitivities to various experimental techniques. A part of this problem is already inherent in the available experimental techniques. Some of these techniques, such as mass spectroscopy, Morjbauer spectroscopy, neutron activation analysis (NAA), and extended X-ray absorption fine structure spectroscopy (EXAFS), identify the chemical nature of an impurity in a host material and measure its total content but in general do not distinguish between various configurations (isolated, associated, substitutional, interstitial, charge state, etc.) of the defect species. Others, such as electron paramagnetic resonance (EPR) and electron nuclear double resonance (ENDOR), additionally identify different charge states and lattice configurations under certain circumstances. Absorption and luminescence techniques, optically detected magnetic resonance (ODMR), photo-thermal ionization spectroscopy (PTIS),various steady state tech-

4.1 Introduction

niques, such as the Hall effect, or transient techniques based on thermal activation (e.g., deep level transient spectroscopy (DLTS)) all measure transitions between different electronic configurations of deep impurities, but often fail to determine the chemical nature or the constituents of a defect. The best way to overcome the problem is to relate “fingerprint” methods to methods sensitive to the chemical nature of a defect species by applying them to identical semiconductor samples. In this context, hybrid techniques like ODMR (Cavenett, 1981), photo EPR (Godlewski, 1985), spin-dependent emission monitored by DLTS (Chen and Lang, 1983) or radiotracer DLTS (Achtziger, 1996) are of valuable help, because they simultaneously cross-link different responses from a defect configuration. For example, Fig. 4-3 shows the picturesque “zoo” of ionization energies ascribed to iron in silicon (Graff and Pieper, 1981). However, the energy level of interstitial iron was definitely shown to be located at 0.375 eV above the valence band, by simply combining the EPR signal from neutral interstitial iron (for the chemical nature)

-- -

--

a0-

173

with the Hall-effect data (for the ionization energy) on identical samples (Feichtinger et al., 1978). As pointed out above, the behavior of deep impurities is essentially influenced by the local interaction of such a center with the surrounding host atoms. For this interaction, experimental data yield a rather paradoxical picture, which is especially manifested in the 3d transition metals series. 1) An atomically localized model (Ludwig and Woodbury, 1962) is strongly favored by EPR data, showing that the net spin associated with 3d impurities obeys Hund’s rule as in the case of free ions. This model is also supported by the multiplets of internal transitions, which resemble atomic multiplets, with ranges of excitation energies similar to free ions (Kaufmann and Schneider, 1983; Landolt-Bornstein, 1982). 2) A model suggesting delocalized states strongly interacting with the host follows from the fact that there is a substantial reduction in the slope of ionization energies with atomic number, relative to free ions (Fig. 4-4). The occurrence of multilevel ionization resulting from different charge states that are energetically con-

-

-

----- ---- - c -y -=---_---z - --- -- -

uc

0 9 0

I

-

Figure 4-4. Experimental 3d ionization energies of neutral free 3d atoms (a) and energy levels for interstitial 3d donors in silicon (b). (After Feichtinger et al., 1984).

174


fined within the small gap region (of the order of 1 eV) of various semiconductors points to an effective screening mechanism and hence also strongly favors a delocalized model. Only the most striking features of experimental data supporting controversial models have been given here, but the puzzle can readily be outlined in more detail (Zunger, 1986). Our general understanding of the physical nature of deep impurities comes from making experimental efforts with respect to the identification problem and from successfully resolving the above-cited paradoxical behavior of deep centers within electronic-structure theory. Accordingly, after introducing some frequently used graphical representations like level schemes and configurational coordinate diagrams, the central purpose of the present article is to figure out different approaches to the electronic structure problem insofar as this is necessary for global understanding, and to emphasize what various models have in common. The remainder of the contribution is devoted to highlighting special features of selected deep center systems via their electrical, optical or kinetic properties. Examples include excited states (chalcogens in silicon), large-relaxation effects (DX centers in semiconductor alloys), metastability and trends of deep donor-shallow acceptor pairs in silicon, and oxygen-related complexes (thermal donors). Some overlap with present structure models should be achieved from these results although longlasting controversies - as in the case of thermal donors in silicon - and many other problems in deep-level physics can not yet be resolved. Finally, the role of hydrogen in the passivation of shallow acceptors and other point defects is sketched.

Wide gap materials like S i c or GaN are not included in the present article because the electronic structure of defects in these materials cannot be considered as being dominated by covalency effects. Experimental and theoretical results concerning defects in S i c and GaN are therefore discussed separately in Chaps. 11 and 13 of this Volume.

4.2 Deep Centers: Electronic Transitions and Concepts 4.2.1 Ionization at Thermal Equilibrium As mentioned above, deep centers often show several charge states readily accessible to experiments that induce one or more ionizing transitions on the same defect. If thermal equilibrium is maintained at a fixed temperature and pressure, the fraction of centers being in a specified formal charge state relative to an adjacent one is given by (Shockley and Last, 1957)

is the increase in Gibbs free where AG(9/9+1) energy upon ionization and refers to the standard chemical potential for transferring an electron from the Fermi level EF to the bottom of the conduction band. The formal charge state of the defect, q, changes to q+ 1, thus making the defect more positive. Electronic degeneracy has been factorized out from AG and is described by the g factors. Thus AG(4’q+1)can be seen as the standard chemical potential for the ionization process, which is the thermally activated part in a reaction like Do * D++e-. Entropy terms still included in AG(y’q+l)arise from a redistribution of phonons populating

175

4.2 Deep Centers: Electronic Transitions and Concepts

the vibrational levels, because ionization usually weakens the lattice modes (Van Vechten, 1982). At temperatures well above 5OO0C, AG(q/4+1)may drastically change, presumably due to non-harmonic effects in the lattice vibrations, whereby the contribution from entropy to AG(q’q+l) strongly increases (Gilles et al., 1990). Entropy terms may be separated from enthalpy terms according to the well-known relation AG(q/q+l)= m ( 9 / 9 + ’ ) - T AS(4/4+1) (4-4) in oder to define “occupancy-levels” = Ec-AH(q/q+l)and the Fermi level EF = E c - k by relating the ionization enthalpy and the chemical potential to the conduction band edge. With these definitions (Baraff et al., 1980), Eq. (4-3) now reads E(q/q+l)

If the weight of degeneracy and entropy is dropped, or there are cases where gq/gq+l G 1, AS(9/@’) E 0, the ratio NdNq+, will be close to unity whenever the Fermi level crosses the occupancy level E(q/q+l). In particular, for amphoteric dopants (defects showing donor and acceptor behavior) and also for “negative U” centers (see later), it is sometimes useful to extend the concept of occupancy levels to two subsequent ionizations. This means determining the Fermi level position for which Nq/Nq+2= 1 in the absence of effects due to degeneracy and entropy. For this purpose, it is necessary to simply replace the numerator in the exponent of Eq. (4-3) by AG(q’q+2) - 2 l ~ =, 2 [[AG‘q/q+2’/2] -A], separate entropy from enthalpy terms [Eq.

(4-4)] and, as before, relate enthalpy [now AZ$9/q+2)/2] and & to the conduction band. This gives

N -Nq+2 -[::2leXp[-

1‘

AS(q/q+?) kB

where E(q’q+2) = (1/2) [ E ( d q + l ) + E ( q + l h + 2 ) 1 is the appropriate occupancy level for two subsequent ionizations. Occupancy levels are obtained experimentally in principle by measuring thermally activated quantities such as DLTS emission rates. These are of the form

where AGn,pis related to the thermally activated process and depends on whether an electron or a hole has been emitted from the defect to the conduction band (AG,) or to the valence band (AGP), respectively. Ionization of the defect therefore occurs according to D9 -+ Dq+’+ e- for electron emission and Dq+’-+ D4+h+ for hole emission. Electronic degeneracy is still contained in the entropy terms of AGn,p,which may be split [Eq. (4-4)] into AGn = m ( 9 / 9 + ’ ) - T &4/4+1)

(4-6 a)

- k B T 1n AG = A H ( S + l / S ) - T A’J’(S+l/S)P

-kB

In

[$1

(4-6b)

Ionization enthalpies are usually obtained from Eq. (4-6) using an Arrhenius plot

176


(log [e,,,(T)/A,,,(T)] over 1/T] and Eqs. (4-6a, b). The factor A,,,(T) in Eq. (4-6) is given by An,p(T) = on,,

( v > n , p Nc,v

I

(4-7)

and contains on,,the thermal-capture cross section for electrons (n) or holes (p), the mean thermal velocity (v),,,, and the effective density of states Nc,v at either the bottom of the conduction band (c) or the top of - T’12 and the balence band (v). Since (u),,, N c,v - T3I2,the temperature dependence of both is (u>,,,N,,, - T 2 :but on,,may also depend on temperature (see Sec. 4.2.2) and, as a rule, has to be measured separately. Equation (4-3) can also be given in terms of hole emission to the valence band according to the reaction Dq+l * DY+h+ with ph-AG(Y+’/q)now being the appropriate numerator in the exponent. Relating the standard chemical potential for holes ( p h ) and the ionization enthalpy [AH‘4+1/4’, Eq. (4-6b)I to the valence band edge gives a Fermi level E , = E, + p h and an occupancy level E(q/q+’) = E,+&Y+l/y’ for hole emission. With these definitions, Eq. (4-5) holds for any equilibrium that can be established between the defect and a band at a fixed temperature and pressure. Once occupancy levels are known for a given center, a level scheme can be constructed like that displayed in Fig. 4-5 for interstitial manganese in silicon. Apart from degeneracies and entropies, it can be stated that when the Fermi level is about midway between the two adjacent Mn-” acceptor and Mno’+ donor levels (Fig. 4-5. above), the impurity will be mainly in the neutral charge state. On the other hand, if the Fermi level crosses the Mn+/++level toward the valence band, for instance (Fig. 4-5, below), the double positive charge state of interstitial manganese will be dominant. Details concerning single level and multilevel systems have been extensively treated

0

+ t tt

Uh

<E

I

1

I

I

Eln/+’

El””’

EF

M’””

1

v

Figure 4-5. Level scheme for interstitial manganese in silicon (Si :MnJ. Dashed arrows indicate the thermally activated processes with corresponding ionization enthalpies. Numerical values for the occupancy levels E(q’q+’) are Ec-0.13 eV (Mn;”), Ec-0.46 eV ( M n y ) and E,+0.24 eV (MnF’+) (Czaputa et al., 1983).

within the framework of semiconductor statistics (Blakemore, 1962; Milnes, 1973; Landsberg, 1982).

4.2.2 Franck-Condon Transitions and Relaxation According to the requirement of thermal equilibrium in the energy level scheme defined above, any level involves a transition between two totally relaxed defect configurations. There is, however, another class of transitions that correspond to centers that have been excited or ionized optically. Excitation by incident light may result, for example, in an internal transition between two localized electronic defect states leaving the formal charge state of the center unchanged. If electron-lattice coupling is sufficiently strong, the lattice atoms near the defect site may be rearranged in response to the changed charge distribution at the defect. But the corresponding relaxation process occurs only after the excitation energy has been absorbed by the defect’s


many-electron system, since lattice relaxations are considered to proceed at a rate about three orders of magnitude smaller than that of the involved electronic system. Figure 4-6 shows the relations between optically induced transitions and lattice relaxation in a so-called configurational coordinate diagram, which helps one to keep track of important terms even in complicated processes. In Fig. 4-6, only the two interesting quantum states are shown by their adiabatic potential energy surfaces. The underlying physical idea in such a diagram is the Born-Oppenheimer approximation (Born and Oppenheimer, 1927; Adler, 1982), which separates electronic motion from that of the nuclei or ion cores (if only outer electrons have to be considered for interaction). Adiabatic then refers to an evolution of a system of electrons and nuclei (cores), where electron eigenstates do not change suddenly on ion core dis-

4 €1 \

I

I

I

Q’ Figure 4-6. Franck-Condon transitions for an absorption-emission cycle resulting in a Stokes shift, AEStokes= 2 S A w. S is the Huang-Rhys factor, which is a measure of the number of phonons effectively involved in a transition and therefore also accounts for electron-lattice coupling strength.

177

placements Q but provide the potential energy for ion core motion. The corresponding vibrational states are then harmonic oscillators with phonon energies E,, = (n+ 1/2) tz o.Each electron quantum state is represented by its own parabola derived in principle by expanding total electronic energies to second order in the configurational coordinate Q at the stable positions (Qo and QR in Fig. 4-6). But it is only for linear coupling that parabolas are of the same shape as that found in Fig. 4-6. Beyond linear coupling, spring constants my be altered in an excitation leading to a shift in mean transition energies with temperature (thereby inducing changes in enthalpies and entropies in the case of ionization) and to unsymmetrical line shapes and mean energies for absorption (E,) and emission (Ee), relative to zero-phonon transitions (Ezp). Mean transition energies corresponding to the most intense lines or to the center of a smoothed-out line shape (which depends on the Huang-Rhys factors S and is Gaussian in the high-coupling and high-temperature limit) depend on the overlap of vibrational wave functions in the initial and final states (Stoneham, 1975, Bourgoin and Lannoo, 1983). According to the Franck-Condon principle, most probable absorption transitions occur between states centered at configuration Qo (Point 1 in Fig. 4-6) for the ground state and states which have their wave functions concentrated near the classical turning points of the same configuration for the excited states (Point 2). When the system has relaxed non-radiatively by phonon emission to the stable configuration at QR, the situation is reversed. Emission of light (luminescence) then proceeds again at constant configuration (Points 3 and 4) followed by non-radiative relaxation of the electronic ground state to the Qo configuration thus terminating the cycle described by

178


Fig. 4-6. This simple picture will work surprisingly well in relating energy shifts, line shapes, and spatial extensions of wave functions to electron-phonon interactions, as long as non-degenerate electronic states can be coupled linearly to phonons of average frequency w by a single representative lattice mode Q. The reason for this success emerges from the fact that strong restrictions exist on mean square displacements and mean square velocities of atoms in a harmonic solid irrespective of strongly temperature-dependent details in lattice dynamics (Johnson and Kassman, 1969; Housely and Hess, 1966). Absorption of light can cause the photoionization of a deep center. The only difference to the charge-state-conserving excitations cited above lies in the fact that either the initial states or the final states are not localized near the defect. Thus, from experimental absorption studies of optical ionizations, thresholds as well as spectra may be obtained in terms of optical cross sections that can be fitted using as parameters the Franck-Condon shift S A w, the spatial extension of the wave function associated with the localized level, and the respective oscillator strengths of transitions toward different extrema of a given band structure (e.g., Bourgoin and Lannoo, 1983). As mentioned before, in the occurrence of a Franck-Condon shift, the equilibrium position of the lattice embedding a defect may vary with its different formal charge states, depending on electron-lattice coupling. Again, such a situation is closely tracked by a configuration diagram. Figure 4-7 depicts a hypothetical donor-type defect which undergoes a distortion i n a generalized configurational mode Q, when in its neutral state. In order to include the band gap, the defect in its positive charge state and the completely filled valence band are

.D'+e'+h'

1 I

'R

1 I

,

3

Figure 4-7. Configurational coordinate model for a deep level. (After Watkins, 1983). For the level scheme, see text.

chosen as an undistorted reference configuration. Therefore, three adiabatic energy surfaces are accounted for, whereby the two centered at Q = 0 are separated by the total energy needed for the creation of an electron-hole pair. This energy does not depend on Q since the process has practically no influence on the configuration of the atoms near the defect. The arrows at the indicate intersection points (marked on,o,,) capture and emission processes which have their physical origin in nonadiabatic transitions left out of the Born-Oppenheimer approximation. Transitions between two vibronic states that differ in electronic energy but have the same total energy are very important in non-radiative trapping and recombination processes. E, in Fig. 4-6 denotes the barrier height by which capture rates appear thermally activated. If lattice relaxation occurs, thermal capture cross sections (Eqs. (4-6), (4-7)) become temperature dependent (e.g., Bourgoin and Lannoo, 1983). In the high-temperature limit, o n , p takes the simple exponential form ( on,p - exp (-E,lkT)). The shifted parabola at QR represents the deep center in its distorted neutral charge state after having trapped an electron from the conduction band.


One may deduce a level scheme for comparison in three ways according to the previous section: (a) ionization via thermal activation, (b) calculation of the transition energy within a rigid, undistorted lattice, and (c) ionization via the absorption of light. It is clear from Fig. 4-7 that the position of the electrical level is given correctly only by the difference in total energies of the stable configurations Do+h+ and D++h++e-, with energies taken at Q = QR and Q = 0, respectively. If relaxation effects are neglected, the ionization energy can be calculated from the difference in electron energies of the neutral and positive charge state of the deep center, both total energies being taken from the undistorted lattice configuration at Q = 0. Finally, if the ionization energy has been obtained from an optical experiment, the initial state will be defined by the relaxed configuration of the neutral defect state at Q = QR while the final state is reached by a Franck-Condon transition at constant configuration. The three ionization energies from Fig. 4-7 therefore appear in a relation Ecalc<Eth= AH(9/9+"< Eopt.It should be noted, however, that by the definition of any chemical potential

179

Zero phonon energies obtained by fitting the spectral dependence of optical cross sections ao(Ao)to an appropriate theoretical model (e.g., Lucovsky, 1965), or by extrapolating cross section data to the optical threshold, can be directly compared to the thermal activation energies Ahr(9/9+1). But this is only practicable if the cross sections are almost purely electronic, which applies to cases where defect-lattice coupling can be ignored (e.g., Schmalz et al., 1994). Although quantitative insight requires appropriate line-shape fitting, weakcoupling conditions (with a Huang-Rhys factor S = 1) at least allow estimation of Et, and Eoptbecause the phonon-dependent line shape of aO(Aw)manifests itself mainly in the "phonon tail" near the photo-threshold, and unfolds to a certain degree from the electronic part of the optical cross section (e.g., Mayo and Lowney, 1986). This is no longer true in the strong coupling regime (S + 1), where accurate information on optical and thermal ionization energies rests on fitting the parameters E,,, S, and A w to the optical line shape, which is dominated by the lattice-relaxation Gaussian. From these parameters, the optical ionization energy, Eopt,may be obtained as the sum of &, and the Franck-Condon shift S A 0 (Legros et al., 1987; Lang, 1986; Sec. 4.4.2.1). Optical cross sections ao(Ao)are usually derived experimentally from optical emission rates

eo = a0@(hw) (4-9) and its application to an optical ionization process (here N denotes the number of where @ ( A o)is the photon flux applied. ionizations), the crystal volume is kept The dependence of ionization energies on constant by virtue of the Franck-Condon symmetric (breathing mode) or symmetryprinciple, whereas entropy S is conserved at lowering distortions (e.g., Jahn-Teller disthe change of internal energy E if no photortions, induced by electronically degenernons are involved in the transition. Thereate defect states with partial occupation (see fore, at low temperatures and when ionizaChapter 3; Sturge, 1967)) may result in a tion entropies are small, [ A H ( ~ / ~ + ~ E) I ~ . reversed ~ level ordering in extreme cases, where, for example, in a multilevel system

180

4 Deep Centers

in Semiconductors

like that shown in Fig. 4-5, the first donor level (O/+) and the acceptor level ( 4 0 ) appear in reversed order. This corresponds to a “negative U’ situation envisioned by Anderson (1975), where the relation UeE = E ( 4 - 1 4 - @/+I) O, the deep level is again host-like, but it shows some bonding character. Pinning to the dangling bond energy of the host occurs at increasing Eimpurity, but this time the deep level enters the gap according to a valence band threshold. No hyper-deep level exists in the case of nominal acceptors. For a large number of sp-bonded impurities that substitute the cation or the anion site and the zincbiende hosts (Si, Ge, Gap, GaAs, InP, InAs, InSb, Alp, AIAs, ZnSe, ZnTe), predictions were made for the relative ordering of ionization energies of al and t, states from the semi-empirical tight binding method. For alloys (e.g., GaAs,,P,), deep trap levels were predicted and experimentally confirmed to follow the pinning energy of the Ga dangling bond or XI (see Fig. 4-12) rather than the points of band edges upon varying the composition (Vogl, 1984).

r,

185

4.3.2.2 Transition Metals In the two-state defect molecule introduced above, the bonding-antibonding repulsion between the impurity s or p orbitals and the a l - and t,-symmetric dangling hybrids of the host were identified to be of crucial importance for the creation of deep levels. If a 3d transition metal atom is put into the dangling bond system of the vacancy, the interaction now involves 3d and 4s orbitals. As stated in Sec. 4.3.1, the 3d orbitals split into a t,-symmetric (d,,-like) triplet and into an e-symmetric (dx2_”,-like) doublet under tetrahedral symmetry. For the d states of the impurity, it is obvious from Fig. 4-9 that only the t2-symmetric d,,-like states can couple effectively to the dangling hybrids of the host, whereas the e-symmetric states retain their atomic character (or, in other words, only the t,-symmetric states can form a-bonding states). Apart from bound states (gap levels), ~ defect-induced ~ ~ y ~ ~ the ~ electronic ~ ~ changes on structure of the host bands (resonances) as well as band states of the pure host that are important for host-impurity interaction, often are given in terms of the “spectral local density of states” (in the form of more or less pronounced peak structures). One may substitute the most important peaks by weighted delta functions in order to get an approximation for the defect molecule picture. Figure 4-13 displays a standard situation as occurring for the heavier elements in the 3d series in silicon according to the results (see next section) of quantitative theories on transition elements in sp3-bonded semiconductors. Again the a,-symmetric (s-like) and the t,-symmetric (p-like) states formed from the dangling bonds and the atomic s and d,, states of the impurity interact according to their relative energetic positions. Thus the antibonding impuritylike al level (resonant with the conduction

186


\

\I

\

\ \

h I'

/ \/ /\

EV

a,\-i

Dangling hybrids

---

\

e+&&-'-

4 -/al_

--2 m 3d

'W 7*

"Molecules"

Atom

Figure 4-13. Simplified defect molecule picture (a further a , resonance from the backbonds of nearest neighbors (Scheffler et al., 1985) is not shown), reflecting qualitatively the interactions of s and d states of substitutional 3d transition metal impurities with the dangling bond system of the host. The single particle levels (spin polarization omitted) are shown as bound states in the gap, the nonbonding e levels and the bonding t2 resonances are populated as an example according to neutral cobalt in silicon. The occupation of the a, resonance, representing the fully occupied a, singlet of the dangling bond system of the vacancy, does not change upon the modeled interaction and therefore is not marked separately. (After Beeler et al., 1990).

band) and the antibonding host-like t2 state constituting the deep level in the gap are created. The resonant a , level in the conduction band remains unoccupied; however, in addition to the two electrons residing originally on the t2 gap level of the neutral dangling bond system, four electrons from the impurity are needed for filling the bonding t2 level deep within the valence band. This confirms the findings of Ludwig and Woodbury presented in Sec. 4.3.1 suggesting a transfer of s electrons into the d shell and of four electrons into the sp3 bonds. The nonbonding e level is not markedly affected by host interaction since the vacancy states have practically no e character (which is the actual justification for the simple picture given in Fig. 4- 13).

However, for interstitial 3d impurities the situation is different. An interstitial atom is embedded in a host environment where all bonds are complete. Thus hybridization is expected only with the tails of sp3 hybrid orbitals on the surrounding atoms. If one compares Fig. 4-9 and Fig. 4-10 for the interstitial site, one realizes that the lobes of d,,.-like (t2) orbitals point to the four nearest silicon neighbors, whereas dxly2-like (e) orbitals point to the six next-nearest neighbors. These directions are imposed by T, symmetry for forming a-type bonds between host p states of e, t2 representation, and d states of an impurity at the interstitial site. Local densities of states (LDOS) of the pure silicon crystal, projected onto the interstitial site (Zunger, 1986; Beeler et al., 1990) show the availability of both t2 states from the first (nearest) coordination sphere and e states from the second (next-nearest) coordination sphere. In principle, this is also confirmed by cluster calculations (De Leo et al., 1981) in the form of discrete dlike levels of a Si,,H,, cluster. Figure 4-14

1'

A

\\I

zizz@B%/l/\ zzzEB%B J\+ eA$,, \ 12 pq e - r,1 -a, '\ -7 \-I

i, %,,2

c

Host

Inter oc t ion host-impurity

Atom

Figure 4-14. Simplified interaction model for a transition metal occupying an interstitial site of tetrahedral symmetry. Gap levels are populated according to neutral manganese in silicon.

4.3 Phenomenological Models and Electronic Structure

depicts the interaction between d-like host states (as before approximated by discrete levels) and the atomic d,,,, orbitals of the transition metal at the T, interstitial site. Since symmetric host al states are also available at the interstitial site, by their interaction with the atomic s states a resonant antibonding impurity-like al state is formed within the conduction band and remains unoccupied. Once again, this is in formal agreement with the Ludwig-Woodbury model, suggesting a transfer of s electrons into the d shell for interstitials as well. The p-d hybridization leads to bound states of e and t, symmetry in the gap, showing antibonding character, and to their counterpart in the form of bonding resonances deep within the valence band. Contrary to substitutional 3d atoms, the gap levels must accommodate all valence electrons of the impurity. An interesting feature of the covalently delocalized models presented so far concerns the relative ordering of the e, t, gap levels: For substitutional 3d impurities, the effective crystal field splitting, with the order being t, above e, results from the hybridization of only the t,-symmetric states with dangling bond host states of the same symmetry, in contrast to the e-symmetric impurity states which remain nonbonding and resemble atomic states. For interstitial 3d impurities, both e- and t,-symmetric states couple to the relevant host states, but the host-like, antibonding t, gap level is repelled by antibonding conduction band states of t, character, while the e gap level is not; hence the order is e above t2 (Fig. 4-14).

4.3.3 Transition Metals: Results of Quantitative Calculations As pointed out above, the relative ordering of the crstal field splitting is also ac-

187

counted for by the covalently delocalized model of a deep center. However, many more physical properties are covered by that point of view as has been revealed by quantitative calculations. In order to emphasize the most striking results, in the following sections the most important consequences are divided into several parts which, in reality, are interrelated.

4.3.3.1 Gap Levels and High Spin-Low Spin Ordering The magnitude of the crystal field splitting (AE,,) manifests itself directly in the trends of the lowest excitation energy of a 3d impurity when multiplet corrections are of minor importance and indirectly when competition with the exchange splitting AEExfor spin-up and spin-down states results in a high-spin or low-spin electronic configuration (see Fig. 4-1 1b). In Fig. 4- 15 the e-t, splitting for silicon is shown as calculated from density-functional theory in conjunction with the local-density approximation. Green’s function technique within the frame work of LMTO-ASA (Linearized-Muffin-Tin-Orbital in AtomicSpheres-Approximation) was used for solving the single particle equations (Beeler et al., 1990). Results were obtained for a rigid (unrelaxed) lattice, neglecting spin polarization (Spin-restricted calculation: The occupancy of all spin-up states is assumed to be equal to the occupancy of all corresponding spin-down states). In the main trend along the 3d series, there is also agreement with calculations based on other methods, such as the X, cluster calculations in Si and GaAs for substitutional impurities (Cartling, 1975; Hemstreet, 1980; Fazzio and Leite, 1980), and in Si for interstitial impurities (De Leo et al., 1981), the empirical tight-binding method for interstitials in Si (Pecheur and Toussaint, 1983), and in

Next Page


Cro Mno Feo Coo Nio Cuo

9

Figure 4-15. Gap levels and apparent crystal field splitting through the 3d transition metal series in silicon for neutral substitutional (solid lines) and interstitial (dotted lines) impurities. Z denotes the effective core charge and is equal to the total number of 3d and 4s valence electrons forming the many-electron system of the deep center. The numbers in parentheses give the population of the levels of different symmetry i n the single-panicle picture, neglecting spin polarization. (After Beeler et al., 1990).

local-density Green's function calculations for interstitial and substitutional 3d atoms in silicon (Zunger and Lindefelt, 1982). A critical discussion of various approaches regarding the choice of an appropriate Hamiltonian and the technique of quantitatively calculating states and energy levels that define a deep center was given by Pantelides (1986). For substitutional transition metals, the trends of the e and t2 levels with atomic number can be easily understood by the help of Fig. 4-13. The impurity-like (d-like) levels of either symmetry will reflect the decrease of atomic de , orbital energy with increasing Z , but the' antibonding t2-symmetric states due to hybridization and pinning to the host dangling bond system show

this trend only in a strongly attenuated fashion as compared to the nonbonding e-symmetric states. Thus the one-electron crystal field splitting is largest at the high-2 limit. Interstitial 3d states will behave in a similar manner, but in this case both e and t2 states hybridize with the host. With decreasing atomic number, as the impurity-host interaction increases, the antibonding t,symmetric states are increasingly repelled by antibonding conduction band states (Fig. 4-14) hence the e-t, splitting is largest at the low Z limit (Fig. 4-15). Interstitial copper does exist only in the single positive charge state, since the completely filled atomic d shell leads to full occupancy of the antibonding e, t2 levels, and the impurity-like al level is resonant with the conduction band (Fig. 4-14); the remaining s electron may be bound by the positive Coulomb potential of interstitial copper resulting in shallow donor behavior. By separating anisotropic many-electron effects, Zunger (1 986) has fitted an average crystal field energy A,, to internal transitions (e" t7 + e"'$ where n+m = n'+m') determined experimentally from excitation spectra in 111-V and 11-VI compound semiconductors; the trend in des behaves like that discussed for substitutional 3d elements in the one-electron picture (Fig. 4- 15), at least from the heavier atoms down to manganese. In calculations within a single-particle model all electrons populating a degenerate one-electron level (like the e or t2 states of Fig. 4-15) feel the same average potential. Although average many-electron effects are contained i n the various one-electron approaches by including exchange terms in effective potentials or energy functionals, this symmetrization in the case of partially occupied states does not allow for stabilizing effects of spatial correlation which may arise from the electrons occupying spatial-

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189


ly separated and variationally independent orbitals (like that shown in Fig. 4-9). Highspin states are simply the result of spatial correlation, since alignment of the spins in strongly localized wave functions (for example, in e and t2 orbitals) keeps the electrons apart and minimizes electrostatic repulsion despite the higher energy of the second orbital (e or t2 for interstitial or substitutional 3d states, respectively). The effect of anisotropic contributions (multiplet corrections) to ground state and transition energies in heteropolar semiconductors was demonstrated in calculations of Fazzio et al. (1 984). The spin part of the correlation energy can be included within the one-electron model by carrying out spin-unrestricted calculations, i.e., allowing electrons of different spin to experience a different potential. The available electrons may then be distributed to the resulting levels in all possible ways. From such states, the various multiplet energies can be obained by perturbation theory, taking into account residual electronic interactions (e.g., Sugano et al., 1970). Possible multiplets for transition metals have also been worked out by Hemstreet and Dimmock (1979) in combining results of one-electron calculation with ligand field theory. However, determining the lowest-energy configuration (the ground state multiplet) is often hampered by the fact that orbitally degenerate multiplets may lead to spontaneous symmetry-lowering Jahn-Teller distortions, thereby stabilizing a low-spin configuration against a nondegenerate, high-spin correlated multiplet originally lower in energy. The question as to the final result of such a competition usually has to be settled by experiment, as has been done for the singly negative vacancy in silicon (see Chapter 3 by Watkins). EPR-results on total spin of 3d impurities in most cases are confirmed by spin-polar-

ized calculations with respect to a rigid lattice. This indicates that Jahn-Teller effects are presumably suppressed by substantial exchange splitting of the gap states. But even though lattice relaxation is neglected, such calculations do not predict an overall high-spin tendency throughout the 3d series and its various charge states (Beeler et al., 1985a, Katayama-Yoshida and Zunger, 1985). Fig. 4-16 and Fig. 4-17 show the spin-polarized one-electron levels in the ground state configuration for neutral, substitutional and interstitial, 3d metals in silicon as obtained from local spin density calculations (Beeler et al., 1985a). The spin splitting depends on spin density and the localization of the defect states. As for the localization, the same qualitative arguments that were used in discussing the trends of Fig. 4-15 hold here as well.

15

\

0.5-

,)I'

>

;0 0 -

._

4

-0.5-1.0

-

-1 5 -

-2.0-

11 3/

S 0

0

1/2

1

3/2

Figure 4-16. One-electron energies (spin-unrestricted) of the ground states of neutral substitutional 3d transition metals in silicon. The numbers in parentheses indicate the occupancy of gap levels and resonant states. (After Beeler et al., 1985 a).

4 Deep Centers Vo

Cr'

Mn'

Fe'

in Semiconductors Coo

NI'

cuo

Figure 4-17. One-electron energies (spin-unrestricted) of the ground states of neutral interstitial 3d transition metals in silicon. (After Beeler et al., 1985a).

For substitutional impurities, the nonbonding impurity-like e states remain strongly localized throughout the series; when they enter the gap, thereby becoming partially occupied, the spin splitting is largest. The antibonding host-like t2 states are strongly delocalized at the high Z limit, due to their largely vacancy-like character, and they become increasingly localized with decreasing atomic number as they are shifted upward toward the conduction band. This causes the e-t2 splitting to decrease (see Fig. 4-15) and the e J state is then pushed above the t2? state for Mno and Cro near the center of the series, resulting in a high-spin configuration for both impurities. For Coo, Ni', and Cuo, however, the nearly constant spin splitting (the increasing magnetization is compensated by increasing delocalization i n that series) is markedly overwhelmed by the e-t2 splitting, the latter being largest at high Z. Therefore, from Fig. 4-16 a low-spin configuration is predicted for neutral substitutional Co, Ni, and Cu in silicon. Unfortunately, experimental evidence is lacking for that prediction,

whereas high-spin states for neutral substitutional Cr and Mn are confirmed by EPR (Fig. 4-1 1 a). Similar arguments hold for interstitial 3d impurities. In Fig. 4-1 7 low-spin configurations are indicated at the low- and high-2 limits, and only the transition elements at the center of the series should form highspin states. Essentially, this is caused by the stronger localization of the e states compared to the t, states and the increase in e-t2 splitting with decreasing atomic number (Fig. 4-1 5 ) . Neutral interstitial Ni is nonmagnetic due to fully occupied states. Therefore, its spin-polarized one-electron configuration is equivalent to that in Fig. 4-15. (This is also true for substitutional Fe', where the e levels are completely filled, and the t2 levels are empty.) Since in silicon 3d metals dissolve mainly on interstitial sites, all high-spin states could be identified by EPR (Fig. 4-1 1 a); again no experimental information exists on the predicted low-spin ground states. For Ti and V on interstitial sites, the singly positive (Ti+) (Van Wezep and Ammerlaan, 1985) and the doubly positive charge state (V2') (Fig. 4- 1 1 a) have been identified by EPR, but for these charge states high-spin and low-spin ordering coincide. (This is easily realized by the use of Fig. 4- 17.) Another interesting fact is revealed by the calculations: The local spin density at the impurity site appears more localized than the gap states themselves; this indicates that an essential part of the local magnetization originates from the irnpurityinduced valence band resonances, although they (like core states) do not contribute to the total spin. A substantial part of the spin density in many cases resides outside the impurity sphere as suggested by ENDOR data. For interstitial Fe+ in silicon, experimental evidence indicates that about 80% (Greulich-Weber et al., 1984) or 72% (Van


Wezep et al., 1985) of the spin density resides inside the impurity first neighbor shell: this value is reduced to about 60% for interstitial Ti+ in silicon (Van Wezep et al., 1985). For comparison, theory yields 73% in the former case and 42% in the latter case (Beeler et al., 1985a). All these findings are also consistent with reduced spin-orbit splitting which is obvious from absorption, luminescence, and photo-EPR experiments on internal transitions (Kaufmann and Schneider, 1983; Clerjaud, 1985) and the quenching of the orbital angular momentum in the g values of EPR spectra mentioned earlier. The origin these effects share, is again the host-impurity interaction via p-d hybridization. Beeler and Scheffler ( 1989) have also performed spin-unrestricted self-consistent LMTO Green's function calculations of the electronic structure of 4d transition metals in silicon. Figure 4-18 shows the results for substitutional impurities in the neutral charge state. In principle, all conclusions drawn from the interaction model of Fig.

Figure 4-18. One-electron energies (spin-unrestnctea') of the ground states of neutral 4d transition metals at the tetrahedral substitutional site. (After Beeler and Scheffler, 1989).

191

4- 13 which are applied to explain the trends depicted in Figs. 4-15, 4-16, and 4-17 remain valid. The only difference concerns the delocalization of both the e and t, states, which is substantially stronger in the 4d series than in the 3d series (This is even more true for the 5d shell.) Consequently, from the calculations represented by Fig. 4-18, low-spin ground states are predicted for substitutional 4d ions. Furthermore, at the high Z end, the t, states again resemble pure dangling bond states. This was already found to be true for the heaviest 3d elements, Cu and Ni. This remarkable property, already inherent in the simplest two-state model of Fig. 4-12 by the existence of a dangling bond pinning energy, is exploited in the so-called vacancy model (Watkins, 1983, 1995). According to this model, the electronic structure of the "heavy" substitutional transition elements can be described in terms of a closed noninteracting d" shell, with the remaining electrons residing at the dangling bond orbitals of the vacancy. One may arrive at this picture by steadily increasing the attractive impurity potential and thus lowering the atomic impurity d,,,, orbital in the defect molecule model of Fig. 4-13. Finally, the nonbonding e states and the now also strongly localized impurity-like t, states, both fully occupied, form the practically noninteracting closed-shell configuration. By now, the host-like gap levels oft, symmetry can be seen to have practically pure vacancy character and no d character. (Hemstreet, 1977). For example, Ni-, Pd-, Pt- correspond to a configuration 3d"+V-, 4d"+V-, 5d"+V-, respectively (V: vacancy). Thus a total spin S = 112, observed in EPR studies for the cited elements and charge states (Pd- and Pt- in silicon, Ni- in germanium), can be readily explained. As for V- (see Chap. 3 of this Volume), the

192


degeneracy of the triply occupied t, gap level (see Fig. 4-13 for the neutral vacancy) is lifted by a tetragonal and a superimposed weaker trigonal Jahn-Teller distortion resulting in C,,, symmetry of the defect. The active electronic configuration is now determined by a spin-paired b, symmetric and a singly occupied b , symmetric split-off state (Watkins and Williams, 1995). This characteristic Jahn-Teller induced distortion is manifested in the EPR spectra of Ptand Pd- (Woodbury and Ludwig, 1962). An analogous EPR spectrum for Ni- in silicon has been detected (Vlasenko et al., 1990). These spectra were reproduced and further resolved by ENDOR studies (Van Oosten et al., 1989; Son et al., 1991), and it has been argued that the large anisotropy in the electronic g factor for Pt- points to a nonvanishing effective angular orbital momentum, contrary to what is expected for the vacancy model. Instead, a dihedral electronic structure has been proposed, consisting of an open d9 shell and two electrons bonded to only two silicon atoms, the two other dangling silicon hybrids forming a reconstructed bond. This model was found to be applicable also to Pd- and Ni-, where an open-shell orbital momentum is possibly reflected by strong anisotropy in the nuclear g factors. On the other hand, in their recent study Watkins and Williams ( 1995) again interpreted and analyzed substitutional Ni-, Pd-, Pt-, and also Auo within the framework of the vacancy model and found the large anisotropy of the hyperfine interaction to be consistent with the b , symmetry of the above-mentioned singly occupied orbital.

4.3.3.2 Coulomb Induced Nonlinear Screening and Self-Regulating Response The preceding sections were intended to demonstrate qualitatively why most transi-

tion metals can support bound states in the gap of a covalent semiconductor: Large variations in the energies of the atomic d states affect the gap levels only weakly via bonding-antibonding splitting. The coexistence of various charge states of a given impurity in the semiconductor gap provides another puzzling fact if one bears in mind that energies in subsequent ionizations of the d shell differ by about 20 eV for a free-space atom. For example, for Mn the d6 to d3 ionization energies are 14.2, 33.7, 51.2, and 72.4 eV, respectively (Moore, 1959; Corliss and Sugar, 1977). The counterpart of this energy for a deep impurity is the Mott-Hubbard correlation energy U,, ( U for unrelaxed lattice), which in silicon is of the order of 0.3 eV (Eqs. (4-lo), (4-11) and Fig. 4-5). This small value of U points to a nonlinear screening of the intra-d-shell repulsion, since a linear response would give a Mott-Hubbard U,, reduced only by one order of magnitude. A self-consistent dielectric function, calculated by Zunger and Lindefelt (1 983) for various transition metals in silicon, reveals that the impurity-induced perturbation is screened within an atomic distance. Consequently, a neutral 3d atom in silicon, substitutional or interstitial, already attains local charge neutrality at the central cell boundary and turns out to be very stable against the build-up of an ionic charge upon ionization. This remarkable dynamic aspect of nonlinear screening was first envisioned by Haldane and Anderson (1976), who concluded that several charge states in the gap occur as a result of rehybridization between transition metal d orbitals and host sp orbitals if the gap level occupation changes upon ionization. The details of this selfregulating mechanism have been worked out extensively by Zunger and Lindefelt (1 983), Singh and Zunger (1 9 8 3 , and Zunger (1 986).


Due to the antibonding character and hence rather extended nature of the gap states, their exclusive contribution to the total charge of the impurity-host system cannot effect the localization of the impurity-induced charge perturbation to the central cell. The difference is compensated by a rearrangement of valence band charge in response to the impurity-induced perturbation. Figure 4-19 shows the contributions of gap states and valence band states (resonances) to the net impurity charge. It can be seen that the gap states only at large distances from the impurity site produce all of the center’s net charge (normalized to unity at the neutrality limit). In addition, the valence band resumes its unperturbed density at precisely these distances. Within the central cell region, the screening response from the valence band amounts to a net charge contribution of up to 50% in the case of neutral interstitial Mn in silicon. For substitutional Mn, a “screening overshoot” occurs (Q,,,>l) very close to the impurity site, arising from the strong bonding resonances.

193

Ionizing the impurity means that an electron is removed from a gap level of a neutral center. This relieves the valence band resonances (Fig. 4-13, 4-14) from their Coulomb repulsion. As a consequence, they become lower in energy, with their coupling to the host being weakened. Dehybridization now strengthens the (localized) d character and weakens the (delocalized) p character of the impurityinduced resonances. Consequently, the resonances increasingly localize in the central cell region thus making up for practically all of the charge that has been lost in the ionization process. Inversely, if an electron is added to a gap level, the mechanism just described would lead to an increase in hybridization, where the d content of the resonances markedly decreases, giving them more p character at the same time. The impurity-induced resonances therefore delocalize, and charge leaks out from the central cell, leaving Qnetnearly as close to neutrality as before. Therefore it appears that by this self-regulating response of the im-

Distance (atomic units) (a) (b) Figure 4-19. Decomposition of the net impurity charge e, into gap level contribution Qgapand the valence band configuration Qvs for interstitial (a) and substitutional (b) manganese in silicon. Local charge neutrality is reached at Qnet= 1, since Qnetis normalized by AZ,the difference in the number of valence electrons between impurity and host. In the cases (a) and (b) displayed in the figure, AZ = 7 and A2 = 3, respectively. Distances from the substitutional or interstitial impurity placed at the origin are given in atomic units (1 a.u. & 0.53 A P 0.053 nm). (After Zunger and Lindefelt, 1982).

194


purity-host system, an ionization-induced change of the charge takes place at the ligands rather than the impurity site. This remarkable stability of 3d impurities in semiconductors against the build-up of an ionic charge upon ionization has been confirmed by other calculations (Beeler et al., 1985a, Vogl and Baranowski, 1985) and explains why several charge states of a 3d metal may occur within the rather small energy limits of a semiconductor gap. It is noteworthy, however, that no comparable mechanism exists for the local magnetic moments (Zunger, 1986), so that exchangeinduced screening is not very effective. Apparently, the main effect on the magnetic polarization of valence band resonances is the change in the net spin of the gap states upon ionization, whereas the varying localization of the resonances is of minor importance. Finally, it should be clear that nonlinear Coulomb screening only works in highly covalent semiconductors. It decreases in efficiency with decreasing covalency and vanishes for ionic crystals.

4.3.4 Ionization Energies and Trends 4.3.4.1 Transition Metals in Silicon Ionization energies as well as excitation energies (Sec. 4.2) correspond to differences in the total energy of the initial and final states of a defect and must therefore include all contributions from electronic and lattice relaxation. Nevertheless, in a rigid lattice, one may approximate the ionization energy of, for instance, a deep donor, within the local density formalism by the self-consistent calculation of Slater’s transition state. This means removing half an electron from the one-electron level to be ionized (e.g., from a t2’? gap level) and transferring it into the conduction band. A comparison with

experimentally determined energy levels can then indicate the relative importance of lattice relaxation effects. Figure 4-20 gives a comparison of calculated (Beeler et al., 1985a) and measured energy levels (Graff and Pieper, 1981 a) of interstitial 3d transition metals in silicon. There is excellent agreement in most cases, but some drawbacks are also evident from Fig. 4-20. While theory yields a double donor for chromium, such behavior has been excluded by experiment (Feichtinger and Czaputa, 1981). Theory has failed to reproduce the correct ground states in the high-spin configuration for Crp and Cr;, Apart from reasons originating from peculiarities of the applied formalism, a key to understanding the missing double-donor activity of chromium is the Cr;” transition. In this transition, the electron is taken from the t,L level whereas the next electron for the Cr:’++ ionization step has to come from the e?’-single-particle gap level (see Fig. 4- 17). Lattice relaxation might cause a general lowering of the calculated e?’ level in this case, to give the observed high-spin ground states for Cry and Cr;, If this lowering would amount to about 0.3 eV, the e?’ level would become resonant with the valence band due to electronic relaxation upon transferring one of its two electrons into the conduction band. Thus doubledonor behavior of Cri is suppressed without seriously affecting the single-donor energy. Unfortunately, for interstitial cobalt, the behavior of the eL level as to its sensitivity to lattice relaxation cannot be tested so far, since the very high diffusivity of interstitial cobalt (e.g., Utzig, 1989) in silicon causes the thermal stability of that species to be too low to observe any predicted energy levels. Interstitial copper has no deep levels, as mentioned before, and should be stable only in its positive charge state. This supports a very high diffusion coefficient and the abil-


Conduction band

-

-

-

0.8 >

-

0.6 ._C

c

W

- - - -. 0.2 0.0

-..

,-

-

Tii

Vi

Cr,

[+/++I

t213/2

Mn,

Fei

Coi

Nii

Cui

-

195

Figure 4-20. Calculated (solid lines) and experimental values (dashed lines) acceptor and donor levels for interstitital 3d transition metals in silicon. For each donor or acceptor level the corresponding change in the occupancy of the single-particle state involved is indicated. Calculations include spin polarization (spin-unrestricted theory). (After Beeler et al., 1985a).

L Valence band

ity of copper to form electrically inactive pairs with acceptor dopants (Mesli and Heiser, 1992). Defect reactions of copper and other fast diffusing impurities in silicon have been studied via transient ion drift experiments (Zamouche et a]., 1995). For similar reasons, nickel in silicon should only be present in the neutral charge state due to its closed d shell. The experimental single-donor levels shown in Fig. 4-20 particularly reflect the free-atom 3d ionization energies (Fig. 4-4) including the characteristic exchangeinduced jump (“Hund’s point”) between Mn and Fe (d5 and d6 in the free atom, respectively). If neutral interstitial vanadium (Vp) has spin S = 5/2, the spin-related jump now should occur between the Vo/+and the Cro/+ level and indeed does so. In case V? has spin S = 1/2 (see Fig. 4-17), this characteristic jump should be related to the increase of the spin splitting of the t, level caused by the change from low-spin to high-spin ground states. For substitutional 3d impurities in silicon, with the exception of Mnt/+ (Czaputa et a]., 1985), no energy levels have definitely been identified until now. 3d metals in silicon dissolve mainly on interstitial sites (Weber, 1983) and often can be transferred

to the substitutional site only by codiffusion of a highly soluble and fast-precipitating species such as Cu (for precipitation effects, see Chapter 11 of this Volume). For Mn:‘+ theory (Beeler et al., 1985a) predicts an energy level at Ev+0.49eV, which is in good agreement with the experimental value of Ev+0.39 eV. A few charge states, for example, Cr:, Mn:, Mn;- (Fig. 4-1 l), and Fe: (Muller et al., 1982), have been identified by EPR and were found to have high-spin configuration. With the exception of the trigonal center Fe:, all the other ones show cubic symmetry. According to theory, neither Fe: nor Fe; should be a stable configuration. Therefore, Fe, is predicted to produce no deep level in the gap. But all this is based on the theoretical result that the low-spin configuration for Fe: (S = 0) is the correct ground state. A highspin ground state (S = 2) would support both the existence of Fe: with spin S = 3/2 and a single-donor level. Theory predicts 4d metals to be dissolved in silicon, preferentially on substitutional sites (Beeler and Scheffler, 1989). Calculated total energies show for the group IB ions (Cu-Ag-Au) representing the 3d, 4d, and 5d series that the stability of the substitutional site increases significantly relative to the

196


interstitial site from Cu to Au because of the increasingly delocalized defect states. Experimental information on the energy levels of 4d metals is rather indecisive; only for Rh is there agreement with theory as to the amphoteric character and level positions. Predictions have been made for the double-donor to double-acceptor activity of substitutional Pd in silicon and for singledonor to triple-acceptor activity of substitutional Ag in silicon (Beeler and Scheffler, 1989). But only isolated interstitial silver (A&) has been unambiguously identified from EPR i n silver-doped silicon, whereas other silver related spectra have been assigned to complexes containing one silver atom, like FeAg (Son et al., 1992). Table 4-1 shows amphoteric behavior for Rh (Czaputa, 1989), Pd (Landolt-Bornstein, 1989), and Ag (Baber et al., 1987) (compare also Sec. 4.5). The tabulated levels are assumed to be related to metals at the substitutional site, a fact which often is supported only by the high thermal stability of the species investigated. It seems clear that for the heavier 4d metals and particularly for the 5d series, Jahn-Teller distortions are no longer suppressed by exchange splitting, due to considerable delocalization of the defect states. Symmetry-lowering distortions in the resulting low-spin configurations are most naturally expected for states Table 4-1. Energy levels (experimental) of substitutional 4d and 5d transition metals in silicon. For a more complete list, see Sec. 3.5. Metal

4d

5d

Donor level (eV)

Acceptor level (eV)

Pd A&?

E, - 0.57 E, + 0.33 E, + 0.34

E, - 0.3 1 E, - 0.23 E, - 0.54

Pt Au

E,, E,

+ 0.32 + 0.35

E, - 0.23

Rh

E,

- 0.55

compatible with the above-mentioned vacancy model. In any case, there is no doubt that for the 4d and 5d series, lattice relaxation should be included in a calculation of level energies to meet the experimental values as closely as possible. Fazzio et al. (1985) have studied the theoretical aspects of the substitutional IB series in silicon. They found that these impurities form a two-level, three-chargestate amphoteric system, where both the donor and acceptor transitions emerge from the antibonding t2 gap state (compare Fig. 4- 18 for Ag and Fig. 4-16 for Cu). But concomitant with the substantial delocalization in this system, Jahn-Teller distortions will occur. Therefore, the expected spin states for the neutral impurities will not be those derived from the unrelaxed lattice (e.g., S = 3/2 for Ag in Fig. 4-1 8) but instead will follow from the occupancy of the latticedistorted split-off t2 states. These should give a doubly occupied spin-paired b2-like level and a singly occupied b,-like level to give S = 1/2 (see again the vacancy model given by Watkins and Williams (1 995)). From Table 4-1, an effective Hubbard energy (Eq. 4-10) of about 0.3 eV can be derived for Ag and Au. Fazzio et al. (1985) have compared their calculated values for the Franck-Condon U (Eq. 4-1 1) with the experimental values for Uefi. They found contributions from lattice relaxation of about 0.15 to 0.25 eV for both donor and acceptor transitions, which tend to cancel each other. The situation should be similar in the case of Pd and Pt. For these metals, U,, appears to be increased to about 0.5 eV (Table 4-l), and the negative charge states should have spin S = 1/2, which is in accordance with the experiment. However, there is still the problem of the correct identification of the isolated substitutional site of transition metals of the 4d and 5d series. The isolated, substitutional character of

197


Pt- which spontaneously distorts off center in a (1 1 1) direction has been confirmed by Anderson et al. (1 992) by studying stressinduced shifts of the g tensor, which clearly demonstrates Czvsymmetry according to the vacancy model. The latter has been thoroughly discussed for the case of Pt(Anderson et al., 1992a) in a following paper. Moreover, the defect has been found to reorient even at 2 K, which also strongly supports its isolated character. As to substitutional gold, Watkins et al. (1992) presented a pioneering Zeeman study on the optical transitions to shallow excited states (Sec. 4.3.5.2) of the gold donor and acceptor. There has been a longlasting discussion on whether the donor and the acceptor transitions evolve from the same center or not (e.g., Lang et al., 1980; Ledebo and Wang, 1983; Utzig and Schroter, 1984). This study not only shows that ground-state properties and ionization energies correspond to the same isolated gold species, but also provides a solution to the long outstanding puzzle concerning the missing EPR spectrum of Auo (S = 1/2). While both charge states Auf and Au-, according to the vacancy model, are expected to have no unpaired spin due to an empty or fully occupied b , state, for the partially filled orbital in the Auo case a definite EPR signal has also never been observed. From the Zeeman spectra in the cited study, a peculiar anisotropy in the g tensor could be established, reflected by the experimental values 911 = 2.8 and g1 = 0 (911 is observed with the magnetic field parallel to a (100) distortion direction). For comparison, these values are 911 = 2.1 and gL = 1.4 for platinum. With gL z 0, no transitions (AM= +I, see Fig. 4.8) can be induced by a microwave field. The origin of this large anisotropy in the gold case is thought to result either from a rapid tunneling of the impurity between two equivalent C,, configurations

driven by a dynamical Jahn-Teller effect, or, preferentially, by an increased spin - orbit interaction which competes with the JahnTeller induced off-center distortion. This very interesting issue has been discussed in great detail by Anderson (1991) and again by Watkins and Williams (1995).

4.3.4.2 Compound Semiconductors and Bulk References Transition metals substituting the cation site in compound semiconductors were studied theoretically by spin-polarized calculations within the framework of the semiempirical tight-binding method (Vogl and Baranowski, 1985). Donors, acceptors, and double acceptors for the 111-V compounds GaAs and GaP and for the 11-IV compounds ZnS and ZnSe were predicted to form highspin ground states. Their calculated energy levels were found to fit closely with the existing experimental data. (For example, properties or iron and chromium in 111-V compounds have been discussed very thoroughly by Bishop (1986) and Allen (1986)). In Fig. 4-21, an example is given for the case of GaAs, from which (this time for the acceptors) the free-ion trend for the energy levels can again be seen in its attenuated form. Note that the characteristic spin-related jump between Mn and Fe arises again from effecConduction band

% .-c

.

1.5

a 0.5-

e

\

/-.

t-

Ti

V

Valence band Cr Mn Fe Impurity

Co

Ni

Cu

Figure 4-21. Calculated (dashed line) and experimental (squares) acceptor levels of 3d transition metals in GaAs. (After Vogl and Baranowski, 1985).

198


tive d5 (Mn-) and d6 (Fe-) configurations constituting the initial states for the ionization of acceptors in GaAs. The same will be true for transition metal donors in 11-VI compounds (see Fig. 4-22 below). As long as high-spin states are involved, this jump will depend both on the formal charge states of the substitutional impurity and on the number of valence electrons of the atoms being substituted. In a further step, this trend can be generalized and unified for a whole class of semiconductors. Level energies (ionization enthalpies) can be measured relative to the valence band edge, the changes of which, at least within certain classes of semiconductor materials (e.g., 11-VI or 111-V), are known to be given approximately by the differences in the experimental photothresholds (Harrison, 1980). In Fig. 4-22 experimental valence-band-related energy levels are shown, corrected for each solid by the corresponding photo-threshold (taken relative to ZnS as a reference host). Evidently, this procedure is very successful in clearly

ZnS ( 0.001

0 CdS (+0.2L)

A ZnSe (+0.68)

\

A CdSe (tO.88) 0 CdTe (+ 1.72)

\

fi

4

8'

'a

TI

V

\

$2

e

cui

Sc

Cr

Mn F e

Co

Ni

Cu

1

Figure 4-22. Experimental valence band related donor lebels of the 3d series in 11-VI compounds, corrected by the corresponding photothreshold, A@. For ZnS, taken as a reference system, A@ is assumed to be zero. (After Vogl and Baranowski, 1985).

exhibiting the universal trend through the 3d transition metal series for different semiconductors (Vogl and Baranowski, 1985) and its close relation to that observed for free ions (Fig. 4-4a). Furthermore, this universal trend points to the existence of a certain reference level to which a given transition metal impurity appears pinned if one changes the host crystal. There is experimental evidence that valence-band offsets in heterojunctions are simply the difference in the energy level positions of a given transition metal in the two compounds that form the heterojunction (Langer and Heinrich, 1985; Delerue et al., 1988). From these findings, an internal (bulk) reference level may be established, circumventing the contribution from the semiconductor surface to the (external) photo-threshold energy (Zunger, 1985; Tersoff and Harrison, 1987). Caldas et al. (1984) suggested this reference level to be related to the antibonding cationic t,-symmetric state (vacuum pinning). There are arguments, however, for the average self-energy of cation and anion dangling bonds constituting the reference in the band alignment of heterojunctions (Lefebvre et al., 1987). Both versions continue to be discussed, but for 111-V compounds, the latter average gives the closer pinning and might be more appropriate at least for this class of semiconductors (Langer et al., 1989). Once such a reference is established, and a deep level of a given meta1 is characterized in one semiconductor, the energy levels in other crystals within the same class may be predicted to a certain degree of accuracy. Further general trends have been discovered in isothermal pressure coefficients of donor or acceptor levels. Deep levels as a rule show pressure coefficients two orders of magnitude larger than those of shallow levels and may therefore be used to define

199

4 . 3 Phenomenological Models and Electronic Structure

deep centers experimentally (Jantsch et al., 1983). Nolte et al. (1987) linked pressure coefficients to the deformation potentials of the band edges by choosing deep centers with a vacancy-like level structure as a reference. Finally, pressure coefficients of deep levels were interpreted in terms of an isothermal change of crystal volume upon ionization to obtain a measure of symmetric (breathing mode) distortions around substitutional and interstitial deep centers (Samara and Barnes, 1986, Weider et a]., 1989; Feichtinger and Prescha 1989).

4.3.5 Excited States 4.3.5.1 Internal Transitions In Sec. 4.2.2, interelectronic transitions at a fixed charge state of both the defect and the accompanying electron-phonon interaction have been discussed. For transition metals, such excitations are better known as internal d + d* transitions according to a change of total energy of the many-electron system

ter, although luminescence may be masked by other competing centers or suppressed by strong electron-phonon coupling, leading to effective radiationless recombination. Zunger (1986) has compiled existing data on internal transitions studied for transition metals in various semiconductors. As a rule, only one “oxidation” state of a given species is observed. This state dN is determined by the total number N of “active” electrons (e.g., d7 for Co2+, or d9 for Ni+) and must not be confused with the charge state of an impurity relative to the given host used so far. For example, an “oxidation” state of Co2+will correspond to the formal charge state Co- in 111-V compounds, to Coo in 11-VI compounds when substituting the cation site, and to Co2- in silicon. For the interstitial site, oxidation state and formal charge state coincide. In order to relate internal transitions to ionization, Fig. 4-23 represents an example, combining experimentally observed single donor activity (Fe’”) and the charge state

9

where i j denotes the multiplet representation indices, and e”” t;’, em t; denote the predominant one-electron configurations (Singh and Zunger, 1985). In Eq. (4-14) an excitonic electron-hole contribution is contained, since both the excited electron and remaining hole are still partially bound. Apart from ground states, interconfiguration mixing may be considerable, and therefore the one-electron level description denotes the configuration that mainly contributes to higher multiplet terms. Numerous internal transitions have been observed via absorption and luminescence spectroscopy (e.g., Landolt-Bornstein, 1989). Both can give valuable information on electronic structure and symmetries of a deep cen-

4

.

t3e’

6 ~ ,

( O / + l a 2 . 4 eV

-3.0 e3t3

3A.

\ t4ez

Fe”‘, d5 t3e2 6A,

0

r,

-1.0 ST,

Fez‘. d6 e3t3 ‘E

J = 5/2

Figure 4-23. Energy level diagram, relating experimentally observed internal transitions to the single donor level of Fe in ZnS. The inserts shows the parameters (including spin J = S for the positive formal charge state) of the ground state configurations participing in the ionization (left) and internal excitations (right). (After Zunger, 1986).

200

4 DeeD Centers in Semiconductors

conserving (Fe') transitions into excited states of substitutional Fe in ZnS. For all states involved, the many-electron multiplets are denoted by spin multiplicin the ity (2S+ 1) and orbital symmetry tetrahedral point group. The oxidation state changes from Fe2+to Fe3+ upon ionization. But transitions to higher excited states of a dN system, occurring as resonances in the conduction band, have also been observed (Baranowski and Langer, 1971). The occurrence of a single fixed oxidation state i n the d + d* spectra (a few transitions have been observed additionally from neighboring states (Zunger, 1986)) and the fact that spectra from internal transitions of a given impurity resemble each other in different semiconductors are not very surprising in light of the electronic structure of deep centers discussed in the present contribution. Since the charge state (or the oxidation state) is conserved, the dynamic aspect of screening described in Sec. 4.3.3.2 is lacking. Consequently, transition energies AE'iJ' (Eq. (4- 14)) cover roughly the same range as found for free ions, suggesting a localized electronic structure (Sec. 4.2.1). Nevertheless, whenever an electron is excited from a nonbonding e level to a hybridized t2 level, it is in fact transferred from the impurity site to the ligands. The optical cross section for the Fe-'O transition in GaAs was found to be strikingly smaller than that of similar ionizations for Cr and Cu (Kleverman et al., 1983), reflecting the different spatial extension of the initial states (e for Fe, t2 for Cr and Cu). It is precisely this difference in orbital character that makes it difficult to treat the multiplet effects of transition metals within empirical ligand field theory. This was pointed out by Watanabe and Kamimura (1 989) who critically reviewed earlier attempts to include multiplet effects in first-

(2s+')r

(r)

principle approaches. Figure 4-24 demonstrates the state-of-the-art accuracy of such calculations compared to experimentally detected multiplet structures. Experimental values are taken from Weakliem (1962) and Koidl et al. (1973) for ZnS, Weber et al. (1 980) for Gap, Hennel and Uba (1978) for GaAs and Baranowski et al. (1967) for ZnSe. From the upper 4A2 + 4T, transition a weak chemical trend with the increasing covalency from ZnS to GaAs becomes evident and is qualitatively reproduced by the calculations. Multiplet structures of interstitial 3d transition metals in silicon were calculated combining the X,-cluster method in a spinunrestricted form with the above-cited approach of Hemstreet and Dimmock to account for space- and spin-induced correlations (DeLeo et al., 1982). Experimentally, Kleverman and Tidlund (1997) concluded from the lack of stress splitting in the optical excitation spectrum of the orthorhombic FeIn pair that the observed lines are most likely due to internal d + d* transitions of the perturbed Fetion (Sec. 4.4.3).

Observed

- ZnSe -GaP GaAs

ZnS

Figure 4-24. Calculated and observed multiplet structures of Co2+ (oxidation state) in 111-V compounds. Theoretical values were calculated by Watanabe and Kamimura (1989).


Internal transitions of rare-earth impurities in various semiconductors provide the possibility of applications for light-emitting devices. In this case, the very atomic-like and nearly host independent luminescence spectra arise from the compact open 4f shell which is screened by the outer 5s5p closed shells. An intensively studied system is erbium in silicon, where the prominent 1.54 pm line offers the benefit of a stable light source compatible with silicon technology [see, for example, Michel et al. (1998)l. When embedded in silicon and GaAs, erbium is observed in the Er3+ oxidation state, which is also expected for other host semiconductors, at least for substitutional erbium (Delerue and Lannoo, 1991). For silicon, however, the tetrahedral interstitial site is favored by theory (Needels et al., 1993) and experiment (Przybylinska et al., 1996; Lanzerstorfer et al., 1998). The luminescence of 1.54 ym wavelength comes from an intra 4f (41,3,2 + 41,5,2) transition between the Two lowest levels separated by spin-orbit interaction. While the wavelength is essentially independent of temperature, this is unfortunately not the case for the efficiency of the luminescence, which decreases significantly with increasing temperature. This main obstacle to the application of both photoluminescence (PL) and electroluminescence (EL) results from the complex nature of the energy transfer between host and Er 4f electrons. It is now well accepted and has been directly observed (Tsimperidis et al., 1998) that the excitation of the Er 4f shell proceeds via the recombination of a transient erbium related bound exciton (BE) state, where the electron-hole pairs are generated either optically (PL) or by carrier injection (EL). Therefore, losses of intensity may result from alternative decay channels of the intermediate BE state or via nonradiative backtransfer mechanisms that de-excite the al-

201

ready excited 4f shell, thus restoring the BE state (Palm et al., 1996). Efficiency improvements in conventional host systems have been reached by co-doping with oxygen [e.g., Takahei and Taguchi (19931, but promising results were also obtained with Si-Ge quantum wells (Huda et al, 1998) and erbium implanted wide gap semiconductors like S i c (see Chap. 11 of this Volume). 4.3.5.2 Rydberg-Like States

In the discussion of the basic physics of a deep center in Sec. 4.1 .I and Fig. 4-2 it was mentioned, that the ground state energy of a deep center is almost exclusively determined by the short-range central cell potential. But whenever a neutral deep center is ionized by emitting either an electron into the conduction band or a hole into the valence band, the screened Coulomb potential due to the charged center should, in principle, give rise to a series of shallow excited states for which effective mass theory (EMT) should be applicable. These closely spaced energy levels near the relevant band edges play a role in the so-called cascade capture. In this process, a carrier is first captured in a highly excited state. It then goes down the ladder by emitting one or very few phonons each time. Finally, the carrier may drop from a lowest-lying cascade state (at low temperatures, re-emission from this state is negligible) into the deeplying ground state by multi-phonon emission or by a lightemitting transition. The cascade process was originally developed for shallow centers (Lax, 1960); in theory, the temperature dependence of the process is expressed in the form T-", where the value of n is usually 1 0) of sufficiently high concentration (above the intrinsic electron concentration ni) p may be replaced by m C,. For donor impurities ( m < 0) analogously the electron concentration is given by I m I C,. For dislocation-free materials, considerations similar to those for uncharged species lead to D$ = ( I m I + 1) (DiCfq/Ctq ) ( C,/Ctq )I I * j (5-41) if the supply of Ai+ limits the incorporation rate. The positive sign in the exponent holds for substitutional acceptors and the negative sign for substitutional donors. The factor Im I + 1 accounts for the electric field enhancement (see also Section 5.4). Equation (5-41) holds both for the generalized kickout and the Frank-Turnbull mechanisms and is independent of the charge state of the native point defects. When the diffusion of self-interstitials to the surface limits the incorporation rate of A, a supersaturation of self-interstitials will develop and the effective diffusion coefficient for the A, atoms is given by D$,

= ( 1 m I + 1) (DI CFq(C,"q)/C,"q)

. (C,/C;q)'k-lml-2

(5-42)

When the supply of vacancies from the surface limits the incorporation of A , an undersaturation of vacancies develops and D$;=

(1 m I + 1) (Dv CGq(Ctq)/Ctq) *

(C,/C;q)'k-m

(5-43)

holds. For both Eq. (5-42) and (5-43) the same sign convention holds as for Eq. (5-41). Equations (5-40)-(5-43) reduce to (5-33), (5-35) and (5-36) if all species involved are uncharged.

246

5 Point Defects, Diffusion, and Precipitation

The quantities D,Cfq (Ctq) and Dv CGq (Ciq) refer to the self-diffusion transport coefficients of Ik+ and V k -under the doping conditions C, = CSeq and not necessarily to the intrinsic self-diffusion coefficient. Even for charged species, constant effective diffusivities may be obtained. For example, for singly charged acceptor dopants (rn = I ) , 13+ ( k = 3) or V- ( k = 1) lead to constant effective diffusivities DL$ and DLg), respectively. Since the applicable effective diffusion coefficient may change with the depth of the profile, complicated concentration profiles may result as frequently observed i n 111-V compounds (Tuck, 1988). Examples of foreign atoms diffusing via one of the interstitial-substitutional mechanisms will be discussed for silicon, germanium and GaAs, in Sections 5.6-5.8, respectively. From a basic science point of view. the importance of the interstitial-substitutional diffusion mechanisms derives mainly from the possibility to distinguish whether self-diffusion in a given semiconductor material is limited by vacancies or self-interstitials, which can hardly be conclusively accomplished by any other means. 5.5.4 Recombination-Enhanced

Diffusion In semiconductors, thermally activated diffusion of defects may be enhanced by the transfer of energy associated with the recombination of electrons and holes into the vibrational modes of the defects and their surrounding as first recognized by Weeks et al. ( 1975) and theoretically described by Kimerling (1978) and Bourgoin and Corbett (1978) in the seventies. The presence of a concentration of electrons and holes above their thermal equilibrium values may be induced by optical excitation (Weeks et al., 1975; Chow et al., 19981, by particle irradi-

ation such as electron irradiation (Bourgoin and Corbett, 1978; Watkins, 2000), ion implantation or plasma exposure (Chen et al., 1996), as well as by carrier injection in devices (Uematsu and Wada, 1992) or via the tip of a scanning tunneling microscope (STM) (Lengel et al., 1995). As a result, the effective thermal activation energy for diffusion may be reduced considerably or may even become essentially zero. In the later case, this type of recombination-enhanced diffusion is termed “athermal diffusion”. The most famous example of athermal diffusion interpreted in terms of recombination-enhanced diffusion appears to be the long-range diffusion of radiation-induced silicon self-interstitials at liquid helium temperatures. In this case, discussed in detail by Watkins (2000) in this volume, the electron-hole excitation is thought to be caused by the particle irradiation. Of technological importance is the recombinationenhanced diffusion of defects in devices such as light-emitting diodes, lasers, or bipolar transistors, including heterobipolar transistors in which carrier injection during operation may lead to undesirable movement of defects. This movement may lead e.g. to dislocation-climb and associated “dark line defects” or other defect rearrangements affecting device performance unfavorably. Convincing direct experiments demonstrating recombination-enhanced diffusion have been reported by Chow et al. (1998).By optically induced EPR, these authors measured a single diffusion jump of a zinc interstitial i n ZnSe induced by optical excitation.

5.6 Diffusion in Silicon 5.6.1 General Remarks

Silicon is the most important electronic material presently used and is likely to keep

5.6 Diffusion in Silicon

that position in the future. Diffusion of dopants is one of the important steps in device processing. For sufficiently deep junctions, diffusion is required for generating the desired dopant profile. For the case of submicron devices the tail of the implantation profile is already in the submicron regime, so that diffusion occurring during the necessary annealing out of implantation-induced lattice damage may already be an undesirable effect. Methods such as rapid thermal annealing by flash lamps are investigated to gain a tighter control over the time spent at high temperatures. Still shallower junctions will probably require closely controlled diffusion processes from well-defined sources, as for example, from doped polysilicon used for certain bipolar devices. The interest in dopant diffusion will increase in this context. Historically, borrowing the knowledge from metals that vacancies are the predominant thermal equilibrium native point defects, diffusion processes in Si had been first described also in terms of vacancy-related mechanisms. In 1968 Seeger and Chik suggested that in Si both self-interstitials and vacancies contribute to self- and dopant diffusion processes. The controversy on the dominant native point defects in Si lasted for almost 20 years. Finally, it became generally accepted that both self-interstitials and vacancies have to be taken into account in order to consistently understand self- and most impurity diffusion processes in the 1980s, with the exception of a few (Bourgoin, 1985; Van Vechten, 1980; Van Vechten et al. 1991). The main indications for the involvement of self-interstitials in diffusion processes in Si came from diffusion experiments performed under non-equilibrium native point defect conditions, such as experiments on the influence of surface oxidation or nitridation on dopant diffusion. Investigations of the diffusion properties of atoms

247

such as Au or Pt migrating via an interstitial-substitutional mechanism were also crucial in establishing the role of self-interstitials in self-diffusion in Si. What is still uncertain is the diffusivity and the thermal equilibrium concentration values of selfinterstitials and vacancies, as will be discussed in Section 5.6.6. 5.6.2 Silicon Self-Diffusion

The transport of Si atoms under thermal equilibrium conditions is governed by the uncorrelated self-diffusion coefficient DSD=D, Cfq+Dv CGq

( 5 -44)

As mentioned in Section 5.2, native point defects may exist in several charge states. The observed doping dependence of group 111 and V dopant diffusion (Section 5.6.4) indicates the contributions of neutral, positively charged, negatively and doubly negatively charged native point defects. It is presently not known whether all these charge states occur for both self-interstitials and vacancies. Taking all observed charge states into account we may write D, C$ as

D, Cfq = DIo C;O'+ DI-CIep + Dp-C;8 + D,+ CJ:

(5-45)

An analogous expression holds for vacancies. The quantity Creq comprises the sum of the concentrations of self-interstitials in the various charge states according to (5-46) c q ; =q c ; + c;3+ c;8+Cfiq Therefore, the diffusivity D, is actually an effective diffusion coefficient consisting of an weighted average of the diffusivities in the different charge states according to Eq. (5-7).The same holds analogously for CGq and D, . The most common way to investigate self-diffusion in Si is to measure the diffu-

248


sion of Si tracer atoms in Si. These tracer atoms are Si isotopes which can be distinguished from the usual Si isotopes the crystal consists of by various experimental techniques. The tracer self-diffusion coefficient DT differs slightly from Eq. (5-44) since it contains geometrically defined dimensionless correlation factorsfi andf,,

tcl

-7

... 1200 1300

E

1100

1000

900

I

t 'o-"ra.

'

self - diffusion in Si

u'\ \

DT=fi D, CFq +fv D , Ctq

(5-47)

The vacancy correlation factorf, in the diamond lattice is 0.5. The corresponding quantityfi I 1 depends on the unknown selfinterstitial configuration. Measured results for DT are shown i n Fig. 5-8. Various results for DT which are usually fitted to an expression of the form

DT= Do exp (- Q/kB T)

(5-48)

are given in Table 5-1 in terms of the preexponential factor Do and the activation enthalpy Q. Tracer measurements, including extensions measuring the doping dependence of DT (Frank et al., 1984), do not allow to separate self-interstitial and vacancy contributions to self-diffusion. Such a separation became possible by investigating the diffusion of Au, Pt and Zn in silicon (as described in more detail in the subsequent Section 5.6.3). These experiments allowed a fairly accurate determination of D, CFqbut only a crude estimate of D, CCq derived from a combination of different types of experiments (Tan and Gosele, 1985). The resulting expressions shown in Fig. 5-9 are

D,c,'~=9 . 4 10-'exp ~ (-4.84ev/kB T)m2s-' (5-49)

D, CGq=6~10-~exp(-4.03 eVlk,T)m2 s-' (5-50)

Values of D,Cfq as determined by different groups are also given in Table 5-1 in the form of a pre-exponential factor and an ac-

1

Ghoshlcgore (1966)

@ Fairfleld ond Moslers (1967)

10"~-

@

Moycr, Mehrcr.Ond Maier ( 1 9 7 7 )

Q

Kolinowsk1 and Seguln (1980)

0

Hirvonen and A n l l i l o (1979)

0 Drrnond

ec 01

\P

(19.83)

from stocking foul1 shrlnhage

tivation enthalpy. It is worth noting that D, CFq coincides within experimental error with 112 DT from tracer measurements. The doping dependence of Si self-diffusion (Frank et a]., 1984) allows to conclude that neutral as well as positively and negatively charged point defects are involved in selfdiffusion, but the data are not accurate enough to determine the individual terms of Eq. (5-45) or the analogous expression for vacancies. Since DT as well as D, Cfq and D, Ctq each consist of various terms, their representation in terms of an expression of the type of Eq. (5-48) can only be an approximation holding over a limited temperature range. In Section 5.6.6 we will discuss what is known about the individual factors D,, C,eq,D , and Ctq.


249

Table 5-1. Diffusivities of various elements including self-interstitials and vacancies in silicon fitted to D = D o exp(-Qlk,T).

Q

References

Description of diffusivity

DO 1o4 m2 s-']

[eVl

Si

DT

1800 1200 9000 1460 8 154 20

4.77 4.72 5.13 5.02 4.1 4.65 4.4

Peart, 1966 Ghostagore, 1966 Fairfield & Masters, 1967 Mayer et al., 1977 Hirvonen and Antilla, 1974 Kalinowski and Seguin, 1980 Demond et al., 1983

Si

D, C$

914 320 2000 1400

4.84 4.80 4.94 5.01

Stolwijk et al., 1984 Stolwijk et al., 1988 Hauber et al., 1989 Mantovani et al., 1986

Si

Dv C$q

0.57

4.03

Tan and Gosele, 1985

I

D,

10-5

3.75 x 1 0 - ~ 8 . 6 lo5 ~

0.4 0.13 4.0

Tan and Gosele, 1985 Bronner and Plummer 1985 Taniguchi et al., 1983

0.1 2500 32 1.9 4.4 0.07 560

2.0 4.97 4.25 3.1 0.88 2.44 4.76

Tan and Gosele, 1985 Hettich et al., 1979 Yeh et al., 1968 Newman and Wakefield, 1961 Tipping and Newman, 1987 Mikkelsen, 1986 Bracht et al., 1998

Diffusing species

V Ge Sn CS

Ci 0 Si

DV DS DS DC Di Di D, C$

5.6.3 Interstitial-Substitutional Diffusion: Au, Pt and Zn in Si Both Au and Pt can reduce minority carrier lifetimes in Si because their energy levels are close to the middle of the band gap. They are used in power devices to improve the device frequency behavior. In contrast, Au and to a lesser extent Pt are undesirable contaminants in integrated circuits and hence have to be avoided. For both reasons, the behavior of Au and Pt has been investigated extensively. Zinc is not a technologically important impurity in Si, but scientifically it served as an element with a diffusion behavior in between substitutional dopants and Au and Pt in Si. The indiffusion profiles of both Au and Pt in dislocation-free Si show the concave

profile shape typical for the kickout mechanism (Stolwijk et a]., 1983, 1984; Frank et al., 1984; Hauber et al., 1989; Mantovani et al., 1986). Examples are shown in Figs. 5-10 and 5-1 1 respectively for Au diffusion and for Pt diffusion. From profiles like these and from the measured solubility CSeq of Au, and Pt, in Si, the values of D, CIeq shown in Fig. 5-9 have been determined. Diffusion of Au into thin Si wafers leads to characteristic U-shaped profiles even if the Au has been deposited on one side only. The increase of the Au concentration in the center of the wafer has also been used to determine D,CIeq (Frank et al., 1984). In heavily dislocated Si the dislocations act as efficient sinks for self-interstitials to

250

-


T [“CI 1100

1300

900

(Perret et al., 1989). In highly dislocated material, an erfc-profile develops as expected (Fig. 5- 12). In dislocation-free material only the profile part close to the surface shows the concave shape typical for the kickout diffusion mechanism. This part can be used to determine D, CFq values as indicated in Fig. 5-9. For lower Zn concentrations, a constant diffusivity takes over. The reason for this change-over from one profile type to another is as follows: In contrast to the case of Au, the Di Ceq value determined for Zn is not much higher than D, Cfq so that even in dislocation-free Si only the profile close to the surface is governed by 0;: of Eq. (5-35) which strongly increases with depth. For sufficiently large penetration depths 0;: finally exceeds D&if’and a constant effective diffusivity begins to de-

800

B

OPerret et a1 (1989) Stolwijk et at (1 983)

6

7

8 9 &/ T[K-’l

10

Au in SI

900 O C , I h

Figure 5-9. Comparison of the contributions D,Cfq and D, CGq to the self-diffusion coefficient in Si determined from the diffusion of Au, Pt, Zn and Ni in Si. Full symbols refer to D,Cfq.

\/

keep C, close to Cfq so that the constant effective diffusivity D;Af from Eq. (5-33) governs the diffusion profile (Stolwijk et a]., 1988). Analysis of the resulting erfc-profiles allowed to determine D, Cleq= 6 . 4 1~0-3exp (- 3.93 eVlk, T)m’ s-‘ (5-5 1 )

In Fig. 5-6, D , C ~ q l C(curve ~ q Aut2’)is compared to D , CfqlC:q (curve Au:’’). D,C,eq turns out to be much larger than D , CFq from Eq. (5-49). This is consistent with the observation that Au concentration profiles are governed by 0;: in dislocation-free silicon. Zinc diffusion has also been investigated in highly dislocated and dislocation-free Si

0.1

Frank - Turnbull mechanism

I

\

1983)

Stolwijk

e

\

0.05 . ~~

2

5

10

20

-

50

x[pml

100

Figure 5-10. Experimental Au concentration profile in dislocation-free Si (full circles) compared with predictions of the Frank-Turnbull and the kick-out mechanism (Stolwijk et al., 1983).

5 . 6 Diffusion in Silicon

251

A

IO 22

-

Mantovani e t al. (1986) 10'70

4 ; x[prnl

Figure 5-11. Platinum concentration profiles in dislocation-free Si (Mantovani et al. 1986).

termine the concentration profile, as shown in Fig. 5-12. A detailed analysis shows that the diffusivity in the tail region may be enhanced by the supersaturation of self-interstitials generated by the indiffusion of zinc, leading to an effective diffusivity in the tail region given by

D$ (tail) =D$,(Ci/Ctq)

0

200

I io0 600

x pm

Figure 5-12. Zinc concentration profiles in dislocation-free and highly dislocated Si. In highly dislocated Si the results can be fitted by a complementary error function (full line), in dislocation-free Si the region close to the surface shows a kickout type profile (Perret et al., 1989).

(5-52)

The changeover from a concave to an erfctype profile has also been observed for the diffusion of Au either into very thick Si samples (Huntley and Willoughby, 1973) or for short-time diffusions (Boit et al., 1990) into normal silicon wafers 300 - 800 ym in thickness. The diffusion profile of Au in Si is very sensitive to the presence of dislocations since dislocations may act as sinks for selfinterstitials and thus enhance the local incorporation rate of Au,. Even in dislocationfree Si the self-interstitials created in supersaturation by the indiffusion of Au may agglomerate and form interstitial-type dislocation loops which further absorb self-interstitials and lead to W-shaped instead of the

usual U-shaped profiles in Au-diffused Si wafers (Hauber et al., 1986). A detailed analysis of Au diffusion profiles at 1000°C by Morehead et al. (1983) showed the presence of a small but noticeable vacancy contribution, which is consistent with the conclusion from dopant diffusion experiments that both self-interstitials and vacancies are present under thermal equilibrium conditions, to be discussed in Section 5.6.4. Wilcox et al. (1964) observed that the Au concentration profiles at 700 "C are characterized by a constant diffusivity, which indicates that at this temperature the kickout mechanism is kinetically hampered whereas the Frank-Turnbull mechanism still

252


operates. This appears also to be the case for the incorporation of substitutional nickel (Kitagawa et al., 1982; Frank et al., 1984). Attempts to repeat the 700°C Au diffusion experiments have failed probably because of a much higher background concentration of grown-in vacancies or vacancy cluster present in nowadays much larger diameter silicon crystals. Nevertheless, the 700 "C Wilcox et al. Au data have been used to estimate D, CGq at this temperature, as indicated in Fig. 5-9.

rich, 1984; Shaw, 1973, 1975; Tuck, 1974; Tsai, 1983). We will rather concentrate on the diffusion mechanisms and native point defects involved in dopant diffusion, the effect of the Fermi level on dopant diffusion and on non-equilibrium point defect phenomena induced by high-concentration indiffusion of dopants. The diffusivities D, of all dopants in Si depend on the Fermi level. The experimentally observed doping dependencies may be described in terms of the expression

D,(n)= D:+D:(ni/n)

5.6.4 Dopant Diffusion

+D,(nln1) +D:-(n/niy

5.6.4.1 Fermi Level Effect

which reduces to

Both n- and p-type regions in silicon devices are created by intentional doping with substitutionally dissolved, group V or I11 dopants which act as donors or acceptors, respectively. Technologically most important are the donors As, P and Sb and the acceptors B and to a lesser extent also A1 and Ga. Dopant diffusion has been studied extensively because of its importance in device fabrication. A detailed quantitative understanding of dopant diffusion is also a pre-requisite for accurate and meaningful modeling in numerical process simulation programs. It is not our intention to compile all available data on dopant diffusion in silicon, which may conveniently be found elsewhere (Casey and Pearson, 1975; Fair, 1981b: Ghandi, 1983: Hu, 1973; Langhein-

D,(n1) =D:+D:+D;+D;-

(5-54)

for intrinsic conditions n = ni. Depending on the specific dopant, some of the quantities in Eq. (5-54) may be negligibly small. D,(ni) is an exponential function of inverse temperature as shown in Fig. 5-6. Values of these quantities in terms of pre-exponential factors and activation enthalpies are given in Table 5-2. Conflicting results exist on the doping dependence of Sb. The higher diffusivities of all dopants as compared to self-diffusion requires fast moving complexes formed by the dopants and native point defects. The doping dependence of D,(n) is generally explained in terms of the various charge states of the native point defects carrying dopant diffusion as dis-

Table 5-2. Diffusion of various dopants fitted to Eq. (5-53). Each term fitted to Do exp(-Q/k,n; IO" rn's-' and Q values in eV (Fair 1981 a, Ho et al. 1983). Element B P As

Sb

(5-53)

0

D, values in

D;

Qo

00,

Q+

D-

Q-

DE

Q2-

0.037 3.85 0.066 0.214

3.46 3.66 3.44 3.65

0.72 -

3.46

-

-

-

-

4.44 12.0 15.0

4.00 4.05 4.08

44.20

4.37

-

-

-


cussed in Section 5.5.2. Since both selfinterstitials and vacancies can be involved in dopant diffusion each of the terms in Eq. (5-54) in general consists of a self-interstitial and a vacancy-related contribution, e.g.,

D; = D:+

DY'

(5-55)

which follows from Eq. (5-27). D,(n) may also be written in terms of a self-interstitial and a vacancy-related contribution (Hu, 1974),

D,( n )=of, ( n )+ DY ( n )

(5-56)

with

@(n) = Dr+D:(niln) + 0:(n/ni) + D: (n/nil2

(5-57)

and an analogous expression for D t ( n ) . Contrary to a common opinion, the observed doping dependence expressed in Eq. (5-53) just shows that charged point defects are involved in the diffusion process, but nothing can be learned on the relative contributions of self-interstitials and vacancies in the various charge states. Strictly speaking, in contrast to the case of self-diffusion, the doping dependence of dopant diffusion does not necessarily prove the presence of charged native point defects but rather the presence of charged point-defectldopant complexes. In Section 5.6.4.2 we will describe a possibility to determine the relative contribution of self-interstitials and vacancies to dopant diffusion by measuring the effect of non-equilibrium concentrations of native point defects on dopant diffusion.

5.6.4.2 Influence of Surface Reactions In the fabrication of silicon devices, thermal oxidation is a standard process for forming field or gate oxides or for oxides protecting certain device regions from ion implantation. The oxidation process leads to the injection of self-interstitials which can

253

enhance the diffusivity of dopants using mainly self-interstitials as diffusion vehicles or retard diffusion of dopants which diffuse mainly via a vacancy mechanism. Oxidation-enhanced diffusion (OED) has been observed for the dopants B, AI, Ga, P and As and oxidation-retarted diffusion (ORD) for Sb (Fahey et al., 1989a; Frank et al., 1984; Tan and Gosele, 1985). The influence of surface oxidation on dopant diffusion is schematically shown in Fig. 5-13. The retarded diffusion of Sb is explained in terms of the recombination reaction (5-10) which, in the presence of a self-interstitial supersaturation, leads to a vacancy undersaturation. The oxidation-induced self-interstitials may also nucleate and form interstitialtype dislocation loops on (1 11) planes containing a stacking fault and are therefore termed oxidation-induced stacking faults (OSF). The growth and shrinkage kinetics of OSF will be dealt with in Section 5.9 covering precipitation phenomena. The physical reason for the point defect injection during surface oxidation is simple and schematically shown in Fig. 5-14. Oxidation occurs by the diffusion of oxygen through the oxide layer to react with the Si crystal atoms at the Si02/Si interface. The oxidation reaction is associated with a volume expansion of about a factor of two which is mostly accommodated by viscoelastic flow of the oxide but partly also by the injection of Si self-interstitials into the Si crystal matrix which leads to a supersaturation of these point defects. The detailed reactions occurring at the interface have been the subject of numerous publications (Tan and Gosele, 1985; Fahey et al., 1989a). Oxidation can also cause vacancy injection provided the oxidation occurs at sufficiently high temperatures (typically 1 150 "C or higher) and the oxide is thick enough. Under these circumstances silicon, probably in the form of S i 0 (Tan and Gosele,

254


Oxidation -influenced diffusion

a)

b)

enhanced diffusion of B,Ga, In,Al,P,As +more 'diffusion vehicles' (self-interstitials, I ) diffusion via I

retbrded diffusion of Sb

-less 'diffusion vehicles' (vacancies, V ) diffusion via v

c* 'cfq 1982; Celler and Trimble, 1988), diffuses from the interface and reacts with oxygen in the oxide away from the interface (Fig. 5- 14). The resulting supersaturation of vacancies associated with an undersaturation of selfinterstitials gives rise to retarded B and P diffusion (Francis and Dobson, 1979) and enhanced antimony diffusion (Tan and Ginsberg, 1983). Thermal nitridation of Si surfaces also causes a supersaturation of vacancies coupled with an undersaturation of self-interstitials, whereas oxynitridation (nitridation of oxides) behaves more like normal oxidation. Silicidation reactions have also been found to inject native point

Figure 5-13. Influence of surface oxidation on dopant diffusion in Si. (a) Cross-section of a Si wafer doped near the surface with B, Ga, In, Al, P, As (left-hand side) or Sb (right-hand side): before oxidation. (b) Same cross-section after surface oxidation indicating enhanced diffusion for B, Ga, In, Al, P, and As. (c) Retarded diffusion for Sb. For details, see text.

defects and to cause enhanced dopant diffusion (Hu, 1987; Fahey et a]., 1989a). A simple quantitative formulation of oxidation- and nitridation-influenced diffusion is based on Eq. (5-56) which changes for perturbed native point-defect concentrations C, and Cv approximately to

w - ' ( n ) = 0:( a )ECIICEq(n>l + m n )[ C V G q ( n ) l

(5-58)

For long enough times and sufficiently high temperatures (e.g., one hour at 1 100°C) local dynamical equilibrium between vacancies and self-interstitial according to Eq.

255


Figure 5-14. Schematic illustration of the injection or absorption of native point defects induced by surface oxidation of silicon according to Francis and Dobson (1979) and Tan and Gosele (1982). (a) Thin oxide layer and/or moderate temperature, (b) thick oxide layer and/or high temperature.

(5-1 1) is established and Eq. ( 5 - 5 8 ) may be reformulated in terms of CI/Cfq.Defining a normalized diffusivity enhancement

A?= [Or'(,) -DS(n)]/Ds(n)

(5-59)

a fractional interstitialcy diffusion component @I

( n )=Dg (n>ID, ( n )

9

(5-60)

and a self-interstitial supersaturation ratio SI@)=

[C,-C,eq(n)llC;q(n)

(5-61)

we may rewrite Eq. ( 5 - 5 8 ) in the form Arr(n)=[2@I(n)+@I(n)sI-l]/(l+sI) (5-62) providedEq. (5-1 1) holds. Usually Eq. (5-62) is given for intrinsic conditions and the dependence of @ I on n is not indicated. Equation (5-62) is plotted in Fig. 5-15 for GI values of 0.85, 0.5 and 0.2. The left-hand side of Fig. 5-15, where sI< 0 (associated with a vacancy supersaturation) has been realized by high-temperature oxidation and thermal nitridation of silicon surfaces, as mentioned above. Another

-I

-

1

0

1

2

3

-

4 SI

5

Figure 5-15. Normalized diffusion enhancement AFr versus self-interstitial supersaturation sI=(CI- C:q)/ CFqfor different values of @, (Tan and Gosele, 1985).

possibility to generate a vacancy supersaturation is the oxidation in an HC1 containing atmosphere at sufficiently high temperatures and for sufficiently large HCl contents (Tan and Gosele, 1985; Fair, 1989). As expected, s, 0) and of thermal nitridation for generating a vacancy supersaturation (sI 0 the diffusion is enhanced and for sI 0.5 holds. Based on the largest observed retardation Ar'(min) (which has a negative

256

5 Point Defects, Diffusion, and Precipitation 'P

$1

- - - - - -- ---

0.5.-

e

1100 O C diffusion in 3

-------

---\-A

+

group ID elements group P elements 1 group 61 elements

t 04 0.6

0.7

0.8

\

0.9

1.0

'.

1.1

rs /rs,

value) a lower limit of ed according to

@I

may be estimat-

> 0.5 + 0.5 [ 1 - ( 1 + A.rr(min))2]1" (5-63) Analogously, an upper limit for @, may be estimated for the case when retarded diffusion occurs for s,> 0 and enhanced diffusion for s,cO, as in the case of Sb. A different procedure is required for elements with @, values close to 0.5. such as As. In Fig. 5-16 values of @ I at 1100°C are shown as a function of the atomic radius rs of the various dopants for intrinsic doping conditions. Both the charge state (group I11 or V dopants) and the atomic size influence O I . @, has a tendency to increase with increasing temperature. Oxidation and nitridation experiments and extrinsic conditions indicate a decreasing value of @, for P with increasing n-doping (see also Section 5.6.4.3), but both P and B still remain dominated by self-interstitials ( @ I (n)>0.5).Inconsistencies in the determination of @ I by oxidation and nitridation experiments (Fahey et al., 1989a) have led to speculations concerning the validity of the basic starting equation (5-58), and to more detailed approaches incorporating a diffusion contribution by the concerted exchange mechanism or the Frank-Turnbull mechanism (Cowern, 1988). @I

Sn

'.$ . q

1.2

Figure 5-16. Interstitial-related fractional diffusion component 0,for group Ill, IV and V elements versus their atomic radius in units of the atomic radius rsi of silicon. The values for carbon and tin are expected from theoretical considerations and limited experimental results.

5.6.4.3 Dopant-Diffusion-Induced Nonequilibrium Effects Nonequilibrium concentrations of native point defects may be induced not only by various surface reactions, as discussed in the previous section, but also by the indiffusion of some dopants starting from a high surface concentration. These nonequilibrium effects are most pronounced for high concentration P diffusion, but also present for other dopants such as B and to a lesser extent for A1 and Ga. In the case of high-concentration indiffusion of P the non-equilibrium concentrations of native point defects lead to a number of phenomena which had initially been labeled anomalous (Willoughby, 1981) before a detailed understanding of these phenomena was arrived at. We just mention the most prominent of these phenomena. Phosphorus indiffusion profiles (Fig. 5- 17) show a tail in which the P diffusivity is much higher (up to a factor of 100 at 900 "C) than expected from isoconcentration studies. In n-p-n transistor structures in which high concentration P is used for the emitter diffusion, the diffusion of the base dopant B below the P diffused region is similarly enhanced. This so-called emitter-push effect is schematically shown in Fig. 5-18a. The diffusion of B, P, or Ga in buried layers many microns away from the P diffused re-

5.6 Diffusion i n Silicon

P, V \

V'

257

electrically inactive P in precipitates

8 PI4

Y

0-

0

1.O

0.5

Figure 5-19. A Schematic P concentration profile (C,) and the normalized native point defect concentrations C,/CF4 and CV/Cveq(Gosele, 1989).

1.5

DISTANCE FROM SURFACE [pn]+

Figure 5-17. Concentration profiles of P diffused into Si at 900" for the times t indicated (Yoshida et al., 1974).

high conc. phosphorus .oxide/ mask

\

-,

boron-doped region

1 si

\ ''

\

\

I

I

\

b)

layer

C)

Figure 5-18. Anomalous diffusion effects induced by high-concentration P diffusion, (a) emitter-push effect of B-doped base region, (b) enhanced diffusion of B, Ga, or As in buried layers and (c) retarded diffusion of Sb in buried layer (Gosele, 1989).

gion is also greatly enhanced (Fig. 5-18b). In contrast, the diffusion of Sb in buried layers is retarded under the same conditions (Fig. 5-18c). The enhanced and retarded diffusion phenomena are analogous to those occurring during surface oxidation. As has also been confirmed by dislocation-climb experiments (Strunk et al., 1979; Nishi and Antoniadis, 1986), all these phenomena are due to a supersaturation of silicon self-interstitials, associated with an undersaturation of vacancies, induced by high-concentration indiffusion of P. The basic features of high concentration P diffusion are schematically shown in Fig. 5-19, which also indicates the presence of electrically neutral precipitates at P concentrations exceeding the solubility limit at the diffusion temperature. A much less pronounced supersaturation of self-interstitials is generated by B starting from a high surface concentration as can be concluded from the B profiles and from the growth of interstitial-type stacking faults induced by B diffusion (Claeys et al., 1978; Morehead and Lever, 1986). Many qualitative and quantitative models have been proposed to explain the phenomena associated with high concentration

258


P diffusion. The earlier models are mostly vacancy based and predict a P-induced vacancy supersaturation (Fair and Tsai, 1977; Yoshida, 1983; Mathiot and Pfister, 1984) which contradict the experimental results obtained in the meantime. Morehead and Lever (1 986) presented a mathematical treatment of high-concentration dopant diffusion which is primarily based on the point defect species dominating the diffusion of the dopant, e.g., self-interstitials for P and B and vacancies for Sb. The concentration of the other native point-defect type is assumed to be determined by the dominating point defect via the local equilibrium condition (see Eq. (5-1 1)). The dopant-induced self-interstitial supersaturation s1may be estimated by the influx of dopants which release part of the self-interstitials involved in their diffusion process. These self-interstitials will diffuse to the surface where it is assumed that CI=Cfq holds, and also into the Si bulk. Finally, a quasi-steady-state supersaturation of self-interstitials will develop for which the dopant-induced flux of injected self-interstitials just cancels the flux of self-interstitials to the surface. The flux of self-interstitials into the Si bulk is considered to be small compared to the flux to the surface. This flux balance may be expressed similarly as in Eq. (5-34) in the case of interstitial-substitutional diffusion as @I

h D,( n) (aC,/&)=-DI(aCIl&)

(5-64)

where 1 Ih I 2 is the electric field enhancement factor and C, the substitutional dopant concentration. With a doping dependence of the dopant diffusivity in the simple form D , ( n)

n'

(5-65)

integration of Eq. (5-64) yields the resulting supersaturation sI of self-interstitials as SI = (CI- cI"q)/c,eq

= [ h @ I D,(n,)

c,l/[(r+1) Dl Cfql(5-66)

For the derivation of Eq. (5-66) the fairly small doping dependence of DICfq has been neglected. In Eq. (5-66) n, is the electron concentration (at the diffusion temperature) and C, the concentration of the electrically active dopants in dimensionless atomic fractions at the surface. For P, y = 2 has to be used. By an analogous equation a vacancy supersaturation may be estimated which may be induced by a dopant diffusing mainly via the vacancy exchange mechanism. Let us briefly discuss the physical meaning of Eq. (5-66). The generation of a high supersaturation of native point defects requires not only a dopant diffusivity which is higher than self-diffusion (which holds for all dopants in Si) but also a sufficiently high dopant solubility. Further simplified, the condition for generating a high supersaturation of native point defects reads

Ds (n,) C,%-.DSD(n)

(5-67)

which is basically the same condition as has been used for the case of generating a nonequilibrium concentration of native point defects by elements diffusing via the interstitial-substitutional mechanism (see Section 5.5.3). In short, diffusion-induced nonequilibrium concentrations of native point defects are generated if the effective flux of indiffusing substitutional atoms (which either consume or generate native point defects) is larger than the flux of migrating host crystal atoms trying to re-establish thermal equilibrium concentrations of the native point defects. We will use this principle again in the context of high-concentration Zn and Be diffusion in GaAs (Section 5.8). In accordance with experimental results, Eq. (5-66) predicts the proper high selfinterstitial supersaturation for P, a factor of up to about eight for B at 900°C and negligible effects for Sb and As. Much more elab-


orate numerical models have recently been proposed for calculating diffusion-induced non-equilibrium point defect phenomena (Orlowski, 1988; Dunham and Wu, 1995).

5.6.4.4 Recombination-Enhanced Diffusion In semiconductors, thermally activated diffusion of defects may be enhanced by the transfer of energy associated with the recombination of electrons and holes into the vibrational modes of the defects and their surrounding as first recognized by Weeks et al. (1 975) and theoretically described by Kimerling (1978) and Bourgoin and Corbett (1978) in the seventies. The presence of a concentration of electrons and holes above their thermal equilibrium values may be induced by optical excitation (Weeks et al., 1975; Chow et al., 1998), by particle irradiation such as electron irradiation (Bourgoin and Corbett, 1978; Watkins, 2000), ion implantation or plasma exposure (Chen et al., 1996), as well as by carrier injection in devices (Uematsu and Wada, 1992) or via the tip of a scanning tunneling microscope (STM) (Lengel et a]., 1995). As a result, the effective thermal activation energy for diffusion may be reduced considerably or may even become essentially zero. In the later case, this type of recombination-enhanced diffusion is termed “athermal diffusion”. The most famous example of athermal diffusion interpreted in terms of recombination-enhanced diffusion appears to be the long-range diffusion of radiation-induced silicon self-interstitials at liquid helium temperatures. In this case, discussed in detail by Watkins (2000) in this volume, the electron-hole excitation is thought to be caused by the particle irradiation. Of technological importance is the recombinationenhanced diffusion of defects in devices such as light-emitting diodes, lasers, or bi-

259

polar transistors, including heterobipolar transistors in which carrier injection during operation may lead to undesirable movement of defects. This movement may lead e.g. to dislocation-climb and associated “dark line defects” or other defect rearrangements affecting device performance unfavorably. Convincing direct experiments demonstrating recombination-enhanced diffusion have been reported by Chow et a]. (1998). By optically induced EPR, these authors measured a single diffusion jump of a zinc interstitial in ZnSe induced by optical excitation. 5.6.5 Diffusion of Carbon

and Other Group IV Elements In Section 5.6.2 we have extensively dealt with self-diffusion of Si. The other group IV elements carbon (C), Ge and Sn are also dissolved substitutionally but knowledge on their diffusion mechanisms is incomplete. The diffusivities of C, Ge and Sn are given in Table 5-1 in terms of pre-exponential factors and activation enhalpies. Ge and Sn diffusion are similarly slow as Si self-diffusion, whereas C diffusion is much faster (Fig. 5-6). Germanium atoms are slightly larger than Si atoms. Oxidation and nitridation experiments show a 0,value of Ge around 0.4 at 1100°C (Fahey et al., 1989b) which is slightly lower than that derived for Si selfdiffusion. Diffusion of the much larger Sn atoms in Si is expected to be almost entirely due to the vacancy exchanged mechanism, similar as for the group V dopant Sb. Consistent with this expectation, a nitridation-induced supersaturation of vacancies increases Sn diffusion (Marioton and Gosele, 1989), but no quantitative determination of @ I is available for Sn. Indiffusion C profiles in Si are error function-shaped. Considering the atomic vol-

260


ume, it can be expected that the diffusion of C atoms, which are much smaller than Si, involves mainly Si self-interstitials. Based on EPR measurements, Watkins and Brower ( 1976) proposed more than 20 years ago that C diffusion is accomplished by a highly mobile carbon-self-interstitial complex (CI) according to

out to the Si surface and hence the Cfq condition is basically maintained, in agreement with experimental observations (Newman and Wakefield, 1961; Rollert et al., 1989). From the C indiffusion data, the solubility of C, is given by (Newman and Wakefield, 1961; Watkins and Brower, 1976; Tipping andNewman, 1987; Rollertetal., 1989)

C,+I

(5-68)

C : q = 4 ~ 1 0 ~exp ~ (-2.3 eVlk,T) m-3 (5-71)

where C, denotes substitutional carbon. This expectation is consistent with the experimental observation that self-interstitials injected by oxidation or high-concentration P indiffusion enhance C diffusion (Ladd and Kalejs, 1986). Equivalently, we may regard C as an i-s impurity, just as Au. That is, to regard the diffusion of C according to (Gosele et al., 1996: Scholz et al., 1998b)

and the diffusion coefficient of D, is given by

C,+I

(CI)

* c,

(5-69)

where C,denotes an interstitial carbon atom. Since whether C, diffusion is actually carried by CI complexes or by C , atoms have not yet been distinguished on a physical basis, and the mathematical descriptions for both cases are identical in form, in the following we will regard C, diffusion as being carried by C, atoms in accordance with the kickout mechanism of the i-s impurities. Under this assumption, diffusion of carbon into silicon for which the substitutional C concentration is at or below the solubility of the substitutional carbon atoms, C:q, the substitutional carbon diffusivity Dtff is given by the effective diffusivity D, C,'qIC,'q where D, is the diffusivity of the fast diffusing C, atoms and Cleq is the solubilities of the C, atoms. Error function type C, indiffusion profiles obtain under indiffusion conditions, because 0,'' Ctq= D,C,eq0 holds. In the case of ionimplantation, both sl> 0 and sv>0 may hold and both quantities will be time dependent. In the case of diffusion-induced nonequilibrium point defects, the presence of dislocations will allow local equilibrium between intrinsic point defects to establish in the two sublattices according to Eq. (5-13). In this way, a large supersaturation of I,, in the Ga sublattice may lead to an undersaturation of I,, or a supersaturation of V,, in the As sublattice.

5.8.3 Arsenic Self-Diffusion and Superlattice Disordering Because there is only one stable As isotope, 75As, As self-diffusion in GaAs cannot be studied using stable As isotopes. In intrinsic GaAs, however, three arsenic selfdiffusion studies have been conducted using radioactive tracers (Goldstein, 196 1; Palfrey et al., 1983; Bosker et al., 1998). In one experiment (Palfrey et al., 1983), the As, pressure dependence of As self-diffusion indicated that As vacancies may be the responsible native point defect species. This is, however, in qualitative contradiction to the conclusion reached recently from a large number of studies involving As atoms and other group V and VI elements that the responsible native point defect species should be As self-interstitials. The latter studies include: (i) As-Sb and As-P interdiffusion in intrinsic GaAs/GaSb,As,, and GaAs/GaP,As,, type superlattices for which x is small so as to avoid a large lattice mismatch (Egger et al., 1997; Schultz et al., 1998; Scholz et al., 1998b); (ii) P and Sb indiffusion into GaAs under appropriate P and As pressures so as to avoid extended

temperature["C] 1200 1100 1000 900

800

. , . , . , . -**, 1 0.70 0.75 0.80 0.85 0.90 0.95

t . , . ,

1/T [IOOO/Kl

Figure 5-30. Data on As self-diffusion coefficient obtained using radioactive As tracers (open squares), the group V elements N, P, and Sb and the group VI donor S (filled symbols). The dashed fitting line is given by Eq. (5-90); the solid line is a better overall fit (Scholz et al., 1998b).

defect formation which leads to complications (Egger et al., 1997; Schultz et al., 1998; Scholz et al., 1998b); (iii) an extensive analysis of the S indiffusion data in GaAs (Uematsu et al., 1995); (iv) outdiffusion of N from GaAs (Bosker et al., 1998). A plot of the relevant data is shown in 5-30. From Fig. 5-30, the lower limit of the As self-diffusion coefficient, assigned to be due to the As self-interstitial contribution, is given by

Da,( n , 1atm) = 6 x 10exp

- 4.8 eV

(5-90)

For P-As and Sb-As interdiffusion, as well as indiffusion cases (Egger et al., 1997; Schultz et al., 1998; Scholz et al., 1998b), the profiles are error function shaped. With P and Sb assumed to be interstitial-substitutional elements, such diffusion profiles are described by an effective diffusivity of the type (5-91)

274


under native point defect equilibrium conditions, which are satisfied by either the kickout reaction (5-30) involving As selfinterstitials or by the dissociative reaction (5-3 I ) involving As vacancies. The conclusion that As self-interstitials are the responsible species is reached for this group of experiments, because the diffusion rate increases upon increasing the ambient As vapor pressure. Arsenic self-interstitials should be the responsible species in the N outdiffusion experiments (Bosker et al., 1998) because the N profile is typical of that due to the kickout mechanism reaction (5-30)under the conditions of self-interstitial undersaturation, which are qualitatively different from those obtainable from the dissociative reaction (5-31 ). Arsenic self-interstitials should also be the responsible species in the S indiffusion experiments, because the S profile (Uematsu et al., 1995) is typical of that due to the kickout mechanism reaction (5-30)under the conditions of selfinterstitial supersaturation, which are also qualitatively different from those obtained from the dissociative reaction (5-31). It is seen from Fig. 5-30 that the available As self-diffusion data lie close to those deduced from the P, Sb, N, and S studies, and it may thus be inferred that As self-diffusion has a component contributed by the As self-interstitials. There are yet no doping dependence studies using the isoelectronic group V elements N. P, and Sb, and hence the charge nature of the involved As self-interstitials has not yet been determined. However, S is a group VI donor occupying the As sublattice sites. In analyzing S indiffusion (Uematsu et al., 1995), it was necessary to assume that neutral As self-interstitial species were involved, which are therefore the most likely species responsible for As self-diffusion. There is also a study on the disordering of GaAs/Al,Ga,-.,As superlattices by the

group IV acceptor species C (You et al., 1993b) which occupy the As sublattice sites. While no information has been obtained from this study on As diffusivity, satisfactory descriptions of the C diffusion profiles themselves were also obtained with the use of the kickout reaction (5-30) involving neutral As self-interstitials. This lends further support to the interpretation that neutral As self-interstitials are responsible for As self-diffusion.

5.8.4 Impurity Diffusion in Gallium Arsenide 5.8.4.1 Silicon Diffusion

For GaAs the main n-type dopant is Si. It is an amphoteric dopant mainly dissolved on the Ga sublattice, but shows a high degree of self-compensation at high concentrations due to an increased solubility on the As sublattice. The apparent concentration dependence of Si diffusion has been modeled by a variety of mechanisms. Greiner and Gibbons (1985) proposed that Si diffusion is predominantly camed by SiAs-SiGapairs. Kavanagh et al. (1988) assumed that the concentration dependence is due to a depth-dependent vacancy concentration generated by an %/As type capping layer. Tan and Gosele (1988b), Yu et al. (1989), and Deppe and Holonyak ( 1988) suggested that silicon diffusion is dominated by negatively charged Ga vacancies, and that its apparent concentration dependence is actually a Fermi level effect. Results of Si diffusion into n-type (Sn-doped) GaAs confirm the Fermi level effect andcontradict the Greiner- Gibbons pair-diffusion model. Deppe and Holonyak (1 988) suggested a charge state of - 1 for the Ga vacancy. Yu et al. (1989) have mainly used V& to fit the Si indiffusion profiles, which is consistent with the species dominating superlattice disordering (Sec. 5.8.2).

275

5.8 Diffusion in Gallium Arsenide

In the analysis of Yu et al. (1989), the diffusivity of the Si donor species Si&, is shown to satisfy 3

k1 -

(5-92)

.--,

to-=

which indicates that V& governs the diffusion of Si;,. In Eq. (5-92), the quantity Dsi (ni) is the Si& diffusivity under intrinsic conditions, identified to be

E

10

&i

( n )= Dsi (ni 1

-d?

-

-18

e-

\

-

---.

-

0.7

for obtaining satisfactory fits to the experimental data of Greiner and Gibbons (1985) and of Kavanagh et al. (1988). In a Si outdiffusion experiment, You et al. (1993a) found that the Si profiles also satisfy Eq. (5-92), but with the needed DSi(ni) values given by Dsi (ni , 1 atm) =

Qi (4, Ga-rich) =

=9.18x104 exp

(-

(5-94 b) 5.25 eV)

2,

s-l

kB

respectively for experiments conducted under As-rich and Ga-rich ambient conditions. The Dsi (ni) expressed by Eq. (5-94) are larger than those of Eq. (5-93) by many orders of magnitude at temperatures above - 800 "C (Fig. 5-3l), indicating the presence of an undersaturation and a supersaturation of V& respectively under the Si in- and outdiffusion conditions (You et al., 1993a). For the indiffusion case, the starting GaAs crystal contains V& and the neutral Ga vacancies V ,: to the thermal equilibrium concentrations of those of the intrinsic material. Upon indiffusion of Si atoms, V& (and hence also Vg,) become undersaturated rel-

indiffusion outdiffusion

0.8

\ \ \

0.9

1 .o

i03/T (K-')

Figure 5-31. Comparison of the intrinsic Si& diffusivities under indiffusion conditions (Yu et al., 1989) and under outdiffusion conditions (You et al., 1993 a).

ative to the thermal equilibrium V& concentration values appropriate for the n-doping conditions, which can only be alleviated via inflow of Vi; from the interface of the Si source material and the GaAs crystal. It appears that the flux of V& flowing into the GaAs crystal is limited by the interface region structural and electrical behavior, which is not sufficiently effective. The reverse analogy holds for the Si outdiffusion case. Since Vi; diffusion should be much faster than that of the S& atoms, in either case there should be no substantial spatial variations in the distribution of the V;, species, while the spatial distribution of Vi; follows the local n3 value.

5.8.4.2 Interstitial-Substitutional

Species The group IV element carbon (C) occupies the As sublattice sites in GaAs to constitute a shallow acceptor species, designat-

276

5 Point Defects, Diffusion, and Precipitation T ("C) 1000

900

)

700

800

10 -17

exp( - 4.47 eV m*s-'

You el al. (1993b)

H As-rich

1 Gerich

=

10

which fits satisfactorily some available data (Jamel and Goodhew, 1993; You et al., 1993b). The values of Eq. (5-95) are shown in Fig. 5-32. In the work of You et al. (1993 b) the C; diffusivity data were obtained by the individual fittings of C; profiles, which are not quite error function shaped. In order to fit these profiles well, it was necessary to use the kickout reaction

-18

-.-

d

E z

L

.-

2 U

lo

-21

i 5

1

0 Saitoet at (1988) -22

A

Cunningham el al (1989)

\ \

~

,o

\\, \

I

0

Chiuetal(1991)

*

Hbfletelal (1992)

0

Jamal e1 al (1994) I

,

I

\ \

c, +I;,

\ \

~

-25

0.7

0.8

0.9

1.o

1,l

1 0 3 / T (K-')

Figure 5-32. Available carbon diffusivity data and fittings in GaAs. (You et ai., 1993 b).

ed as C; to emphasize that it is most likely a interstitial -substitutional species. Grownin during MBE crystal growth, C; reaches high solubilities (Konagai et al., 1989) and diffuses slow (Cunningham et al., 1989), which are attractive features when compared to the main p-type dopants Zn and Be in GaAs. The measured C; diffusivity values of a few groups obtained under As-rich annealing conditions (Saito et al., 1988; Cunningham et al., 1989; Chiu et al., 199 1 : Hoffler et al., 1992; Jamel and Goodhew, 1993; You et al., 1993b) are fitted well by the expression

D,( I atm)=4.79 x 10" - 3.13 eV

(5-95b)

kB

(5-95 a)

The corresponding D,values under Ga-rich conditions should therefore be

0

c;

(5-96)

where C; is an interstitial C atom, which is also assumed to be an acceptor, and I:, is a neutral As self-interstitial, together with a carbon precipitation process. Later, Moll et al. (1994) identified the nature of the precipitation process as that of graphite formation. The As self-interstitials are maintained at their thermal equilibrium values during C; diffusion, because of its low diffusivity value. The main p-type dopants in GaAs based devices, Zn and Be, diffuse via an interstitial-substitutional mechanism in GaAs as well as in many other III-V compounds. Although in most papers Zn and Be diffusion has been discussed in terms of the much earlier suggested Frank-Turnbull or Longini mechanism (Casey, 1973; Tuck, 1988), only the kickout mechanism involving Ga self-interstitials is quantitatively consistent with the superlattice disordering results (Sec. 5.8.2) as well as with the Zn diffusion results (Yu et al., 1991 a; Jager et al., 1993). Isoconcentration diffusion of Zn isotopes in GaAs predoped by Zn showed error function profiles (Chang and Pearson, 1964; Ting and Pearson, 197 1 ; Kadhim and Tuck, 1972) with the substitutional Zn diffusivity

5.8 Diffusion in Gallium Arsenide T (“C)

1200 I

900

600

1

Winteler (1971)

*

,

d

E

I

Kadhim 8 Tuck (1 972) Ting & Pearson (1971) Kendall (1968) Casey et al. (1967)

Casey B Panish (1 968)

Chang & Pearso (1964) 0.8

1 .o

1.2

exp!

103/T (K-’)

Figure 5-33. The substitutional Zn diffusivity values under intrinsic and 1 atm As4 pressure conditions (Yu et al., 1991 a).

values of

D,( p , latm) = D,(ni, latm)

[:I

-

(5-97)

for As-rich GaAs and an analogous expression for Ga-rich GaAs. At sufficiently high Zn concentrations, since the GaAs hole concentration p approximately equals the Zn, concentration ( p C,),Eq. (5-97) shows that the responsible native point defect species can only be the doubly positively charged Ga self-interstitials or vacancies, 1% or V z . Under high concentration Zn indiffusion conditions, the GaAs/AI,Ga,,As superlattices disordering rates are tremendously enhanced (Sec. 5.8.2), indicating the presence of a high supersaturation of the responsible point defects. Thus the native point defect species responsible for Zn diffusion, and also for Ga self-diffusion and AI-Ga interdiffusion under p-doping conditions, is 1% and not V z . In the latter case, only an undersaturation of V z can be incurred by Zn indiffusion, which should then

-

retard AI-Ga interdiffusion rates in superlattices, in contradiction to experimental results. In the Zn isoconcentration diffusion experiments, a nonequilibrium 1% concentration is not involved. Similarly, for Zn diffusion to low concentrations below the ni value, a nonequilibrium concentration of 1 2 is also not present, and the Zn diffusivity values may be represented by that under intrinsic conditions, D,(ni). As analyzed by Yu et al. (1991 a), Zn isoconcentration experiments and Zn indiffusion experiments at high concentrations yielded the value range of Ds(ni,1 atm)=1.6x104

o - ~ ~

1

0.6

277

- 2.98 k B TeV

1

m2s-l

(5-98a)

The two analogous expressions for galliumrich materials are respectively 0,( q ,Ga-rich)=l. 18 x lo-’’

(5-99a)

, ( q , Ga-rich)=7.14~10-’

(5-99 b) The values of Eq. (5-98) and the associated data are plotted in Fig. 5-33. The correspondingly deduced 1% contribution to gallium self-diffusion has been included in Eqs. (5-82) and (5-83). Because of the lack of a proper beryllium source for indiffusion studies, and in beryllium outdiffusion studies with beryllium incorporated using MBE or MOCVD methods the beryllium diffusivity is too small, there are no reliable beryllium diffusivity data.

278


Outdiffusion of Zn or Be in GaAs doped to fairly high concentrations during crystal growth but without introducing extended defects is associated with a high 1% undersaturation, leading to Zn or Be outdiffusion rates orders of magnitude smaller than those under indiffusion conditions (Kendal, 1968; Truck and Houghton, 1981; Enquist et al., 1985). Indiffusion of high concentration Zn into GaAs induces an extremely large 1% supersaturation, because the condition

Di CleqSDE: ( p )

( 5 - 100)

holds. As first noted by Winteler (197 l), this 1 2 supersaturation leads to the formation of extended defects. In recent works, three kinds of extended defect have been characterized and their formation process analyzed (Luysberg et al., 1989, 1992; Tan et al., 1991a: Jager et al., 1993): (i) interstitialtype dislocation loops, which degenerate into dislocation tangles in time: (ii) voids; and (iii) Ga precipitates neighboring voids. For diffusing Zn into GaAs in a Ga-rich ambient, a Zn diffused GaAs crystal region with compositions at the allowed Ga-rich boundary shown in Fig. 5-4a is obtained, irrespective of the GaAs starting composition. The fact that the Zn diffused region is indeed rich in Ga is evidenced by the presence of Ga precipitates in the voids (Luysberg et al., 1989, 1992; Jager et al., 1993). Formation of these defects ensures that the Zn indiffusion profile is governed by the thermal equilibrium concentrations of native point defects of the Ga-rich GaAs crystal, and the profile is box-shaped, which reveals the p* (or C:) dependence of the substitutional Zn, D,. Such a profile is shown in Fig. 5-34 together with an illustration of the involved extended defects. It is, however. noted that the crystal is in a highly nonequilibrium state, for two reasons: First, extended defects are generated. Second, the

starting material may not be rich in Ga and hence the crystal will now contain regions with different compositions which is of course a highly nonequilibrium crystal. For diffusing Zn into GaAs in an As-rich ambient, the situation is more complicated. After a sufficient elapse of diffusion time, the crystal surface region becomes As-rich because of the presence of a high ambient As, pressure. But since (5-101) holds in the Zn diffusion front region, it is Ga-rich. Thus the high concentration Zn indiffusion profiles are of a kink-and-tail type resembling those of high concentration P indiffusion profiles in Si, see Fig. 5-35. The kink-and-tail profile develops because the Zn, solubility value in the As-rich and Garich GaAs materials are different (Jager et al., 1993). In the high Zn concentration region the D,(ni) values are those given by Eq. (5-98),while in the tail or Zn diffusion front values are those given by region the D,(ni) Eq. (5-99). These profiles cannot be modeled with a high degree of self-consistency, because the extended defect formation process cannot be modeled without the use of some phenomenological parameters (Yu et al., 1991 a). The evolution of the extended defects, as suggested by Tan et al. (1991 a) and Luysberg et al. (1992), is as follows: (i) to reduce 1% supersaturation, they form interstitial-type dislocation loops containing extra GaAs molecules, with the needed As atoms taken from the surrounding As sites, which generates a VASsupersaturation: (ii) the supersaturated VAS collapses to form voids, each of an initial volume about that of a neighboring Ga precipitate formed from Ga atoms lost their neighboring As atoms to the formation of dislocation loops. The voids will be rapidly filled by subsequently generated Ga self-interstitials due to further Zn indiffusion. For cases of diffusing Zn into

279

5.8 Diffusion in Gallium Arsenide

. 5

J

c

'-

'O'$ 10 17

0

I

' i

40

80

160

120

200

xllkml

b)

I

I

0

20

40 XI

60 80 [VI

100

120

I

Figure 5-34. a) Zn indiffusion profiles obtained at 900 "C under Ga-rich ambient conditions. Squares are the total Zn concentration and crosses are the hole or Zn, concentration. The higher total Zn concentration indicates the formation of Zn containing precipitates caused by the use of a nonequilibrium Zn source material which diffused Zn into GaAs exceeding its solubility at 900°C. b) A schematic diagram indicating the morphologies and distributions of voids (open) and Ga precipitates (filled), also indicated by v [p]. The presence of dislocations is not shown (Jager et. a]., 1993).

Figure 5-35. a) Zn indiffusion profiles obtained at 900°C under As-rich ambient conditions. Squares are the total Zn concentration and crosses are the hole or Zn, concentration. The higher total Zn concentration indicates the formation of Zn-containing precipitates caused by the use of a nonequilibrium Zn source material, which diffused Zn into GaAs exceeding its solubility at 900°C. b) A schematic diagram indicating the morphologies and distributions of voids (open) and Ga precipitates (filled). The presence of dislocations is not shown (Jager et al., 1993).

GaAs in a Ga-rich ambient, the voids contain Ga precipitates throughout the Zn indiffused region, but for cases of diffusing Zn into GaAs in an As-rich ambient, the surface region voids are empty. Chromium is a deep acceptor occupying Ga sites and is used for fabricating semi-insulating GaAs. In GaAs not deliberately doped by a shallow dopant, diffusion of Cr involves no charge effects. Indiffusion profiles of Cr are characterized by a kickout type profile from the crystal surface to a substantial depth and an erfc-type profile deeper in the material near the diffusion front (Tuck, 1988; Deal and Stevenson, 1988). Outdiffusion profiles are characterized by a constant diffusivity, which is much lower than for in-diffusion. The existence of the

two types of profile needs the description of the interstitial -substitutional diffusion mechanism in terms of the kickout mechanism (Eq. 5-30)) and the Franke-Turnbull mechanism (Eq. 5-31)). Tuck (1988) and Deal and Stevenson (1988) have discussed Cr diffusion in terms of the Frank-Turnbull mechanism. The satisfactory treatment of the diffusion behavior of Cr in intrinsic GaAs (Yu et al., 1991b), however, includes to co-existence of Ga vacancies and selfinterstitials, the dependence of CSeq and Cfq on the outside Cr vapor pressure, and a dynamical equilibrium between the native point defects in the Ga and the As sublattice at the crystal surface region. Chromium indiffusion turned out to be governed by the concentration-dependent D7[:vl from Eq.

280


(5-37) in the surface region and by the much faster constant diffusivity D $ , from Eq. (5-52) in the tail region. In the case of outdiffusion, the Cr vapor pressure is so low that, similarly to the case of outdiffusion of Zn, a much lower diffusivity prevails. This slower outdiffusion turned out to be dominated either by the constant vacancy component of D;Tv, or the constant DZY, which can be lower than D7Fv, for low outside chromium vapor pressure. The deduced D, CFq value from Cr indiffusion profiles (Yu et al., 1991 b) has been included in Eq. (5-82). The group VI donor S occupies As sites. With lower surface concentrations, the S indiffusion profiles (Young and Pearson, 1970; Tuck and Powell, 1981; Uematsu et al., 1995) resemble the erfc-function, but a concave shape develops in the surface region for higher concentration cases. The latter cases are indicative of the operation of the kickout mechanism for an interstitial-substitutional impurity. The available S indiffusion profiles have been quantitatively explained (Uematsu et al., 1995) using the kickout mechanism assuming the involvement of the neutral As self-interstitials, I i s . The deduced DL,(ni, 1 atm) values were included in Eq. (5-90). 5.8.5 Comparison to Diffusion in Other III-V Compounds

Gallium arsenide is certainly the one III-V compound in which self- and impurity diffusion processes have been studied most extensively. The available results on self-diffusion in III-V compounds have been summarized by Willoughby ( 1983). The Group I11 and the Group V diffusivities appear to be so close in some compounds that a common defect mechanism involving multiple native point defects appears to be the case, although no definite conclusion

has been reached. There are hardly any experimental results available which would allow conclusions to be drawn on the type and charge states of the native point defects involved in self-diffusion processes. Zinc is an important p-type dopant also for other III-V compounds, and its diffusion behavior appears to be governed by an interstitial -substitutional mechanism as well. No information is available on whether the FrankTurnbull mechanism or the kickout mechanism is operating. It is to be expected that dopant diffusion induced superlattice disordering may rapidly advance our understanding of diffusion mechanisms in other III-V compounds similarly as has been accomplished in GaAs. The state of understanding of diffusion mechanisms in II-VI compounds has been discussed by Shaw (1 988).

5.9 Agglomeration and Precipitation In semiconductors, agglomeration and precipitation of an impurity or native point defect species are general phenomena which exist in an excess of an appropriate thermal equilibrium concentration or solubility. Due to the supersaturation of native point defects developed during cooling, swirl defects form in Si during crystal growth. The solubility of an impurity species is defined by the thermal equilibrium coexistence of the semiconductor and a unique compound phase of material composed of the impurity atoms and elements of the semiconductor. For practical reasons, however, a nonequilibrium source material is usually used to indiffuse dopants into the semiconductor. Thus high concentration P indiffusion into Si is associated with the formation of Sip precipitates near the surface region, and high concentration Zn indiffusion into GaAs is also associated with the formation of Zn

5.9 Agglomeration and Precipitation

containing precipitates. In this sections we discuss the agglomeration phenomena of native point defects and impurity precipitation phenomena. For the latter category, those associated with the use of nonequilibrium diffusion source matrials will not be included for they appear to be relatively trivial cases.

5.9.1 Agglomerates of Native Point Defects in Silicon Nonequilibrium concentrations of native point defects develop in Si during crystal growth, ion-implantation, and surface processes such as oxidation or nitridation. The nonequilibrium native point defects associated with crystal growth may agglomerate to generate various types of so-called swirl defects. A-swirl defects consist of interstitial-type dislocation loops resulting from a supersaturation of Si self-interstitials. Bswirl defects are considered as a precursor of A-swirl defects, probably consisting of three-dimensional agglomerates of selfinterstitials and carbon atoms (deKock, 1981; Fall etal., 1981). Agglomerates ofvacancies have been termed "D-swirl" defects (Abe and Harada, 1983). Voids to sizes of 100 nm have been found in recently available large diameter (30 cm) CZ Si crystals (Kato et a]., 1996; Ueki et al., 1997), which are apparently D-swirl defects grown to large sizes. These voids are supposed to be responsible for low gate break-down voltages in MOSFET devices (Park et al., 1994). The formation of swirl defects results from a supersaturation of Si self-interstitials or vacancies, due to cooling in crystal regions moving away from the crystal-melt interface wherein the native point defects are at their thermal equilibrium values at the Si melting temperature of 1412°C. The formation of swirl defects depends on the growth speed and the temperature gradient in the

-

281

crystal. After the way having been paved by many previous attempts (Voronkov, 1982; Tan and Gosele, 1985; Brown et al., 1994; Habu et al., 1993a, b, c), a seemingly satisfactory quantitative model on the swirl-defect formation process is now available (Sinno et al., 1998). As a function of the crystal growth rate and the temperature gradient, this model fits fairly well the experimentally observed swirl-defect type, size, and distribution. In the model, basically Eqs. (5-49) and (5-50) are used for the Si self-interstitial and vacancy contributions to Si self-diffusion, with the appropriate point defect thermal equilibrium concentration and diffusivity values already discussed in Sec. 5.6.2. An important aspect to note is that the used Si vacancy migration enthalpy is less than 1 eV. In a simplified model describing the void growth process from supersaturated Si vacancies, Plekhanov et al. (1998) also needed to use a Si vacancy migration enthalpy value of less than 1 eV. Due to the complexities involved, a detailed discussion of the swirl-defect formation process appears to be beyond the scope of the present chapter. In the following, we will deal with the much simpler case of the growth or shrinkage of dislocation loops containing a stacking fault on (1 11) planes. Such dislocation loops may be formed by the agglomeration of oxidation-induced self-interstitials, and have been termed oxidation-induced stacking faults (OSFs). These stacking faults may either nucleate at the surface (surface stacking faults) or in the bulk (bulk stacking faults). Approximating the shape of the stacking faults as semicircular at the surface with radius rSF in the bulk, we may write their growth rate as r

- 4 C ; q ~ ~ + & C $ q s vsz] A(5-102)

282


In Eq. (5-102), aeffis a dimensionless factor which can be approximated as about 0.5, ySF(= 0.026 eV atom-') denotes the extrinsic stacking fault energy, A(=6.38~10-" m2)the stacking fault area ~ ~ the per atom, and S Z ( = ~ . O X I O -m3) atomic volume. In the derivation of Eq. (5-102) it has been assumed that ysFlkBT+l and that the line tension of the Frank partial dislocation surrounding the stacking fault may be neglected i n comparison to the stacking fault energy. The first condition is always fulfilled (e.g., ysFlk, T = 0.2 at 1300 K), the second for r S F 21 ym. The quantities sI and sv denote self-interstitial and vacancy supersaturations, respectively, defined analogously to Eq. (5-61). In an inert atmosphere, native point defect equilibrium is maintained (sI=0, sv = 0) and Eq. (5-102) reduces to

/

\

-

(5-103) which describes a linear shrinkage of stacking faults, as has been observed experimentally (Fair, 1981 a; Frank et al., 1984). From measured data of (drSF/dt),", the uncorrelated self-diffusion coefficient D S Dmay be determined. The results are included in Fig. 5-8. Quantitative information on sI has been extracted from the growth rate (drsF/dt)ox of OSF under oxidation conditions, together with the shrinkage rate (drs,ldt)i, in an inert atmosphere at the same temperature

Equation (5-104) yields for dry oxidation of a ( loo} Si surface at temperatures in the vi-

cinity of 1100°C 9 -114

sI = 6.6 x 10- t

exp

2.53 eV

( 5 - 105)

Kg

(Antoniadis, 1982; Tan and Gosele, 1982, 1985). For ( 1 1 1 ] surfaces, the right-hand side of Eq. (5-105) has to be multiplied by a factor of 0.6-0.7 (Leroy, 1986). For wet oxidation, multiplication factors larger than unity have to be used. The supersaturation ratios sl calculated based on Eq. (5-105) appear to overestimate sI by 20-50%. 5.9.2 Void and Gallium Precipitate

Formation During Zinc Diffusion into GaAs In elemental crystals, a supersaturation of native point defects may be eliminated by the nucleation and growth of dislocation loops. In this way, Czq may be established by dislocation climb processes. As discussed in Sec. 5.3, in compound semiconductors dislocation climb involves point defects in both sublattices. A supersaturation of Ga self-interstitials, as induced by high concentration Zn diffusion into GaAs, can be reduced by dislocation climb processes under the simultaneous generation of As vacancies or the consumption of As self-interstitials. Dislocation climb will stop when local point defect equilibrium according to Eq. (5-13) has been reached (Petroff and Kimerling, 1976; Marioton et al., 1989). Therefore dislocation climb alone does not generally establish the thermal equilibrium concentration of native point defects. Thermal equilibrium concentrations in both sublattices may be reached if the As vacancies generated in the As sublattice via dislocation climb agglomerate and form voids. These voids will be in close contact with Ga precipitates (in a liquid form) of about the same volume. The Ga precipitates form

283

5.9 Agglomeration and Precipitation

Carbon

Oxygen

L Volume increase (foctor 2)

s+

I

9. Sic or C agglomerate Volume reduction (factor 2)

agglomerates if 1 in supersaturotlon co-precipitation I and C

Oi

+

I'B-swirIs''

Figure 5-36.Schematic representation of volume changes during oxygen precipitation (left) and carbon precipitation (right), and the respective role of Si selfinterstitials (Gosele and Ast, 1983).

from those lattice Ga atoms that lost their neighboring As atoms to the dislocation climb process. Subsequently, the voids may act a sinks for more Ga interstitials by continued Zn indiffusion. With the combination of the growth of interstitial-type dislocation loops and of voids neighboring Ga precipitates, supersaturations of I,, will be completely relieved and thermal equilibrium concentrations of native point defects in both sublattices establish, in accordance with those for a GaAs crystal with a composition at the thermodynamically allowed Ga-rich crystal limit. Such a defect structure has in fact been observed in transmission electron microscopy studies of Zn-diffused GaAs (Luysberg et al., 1989).

5.9.3 Precipitation with Volume Changes in Silicon Oxygen and carbon are the main electrically inactive impurities in CZ Si. Oxygen is incorporated from the quartz crucible during the CZ crystal growth process in the form of oxygen interstitials Oi. The concentration Ci of these interstitial oxygen atoms at typical processing temperatures is higher than their solubility Cfq at these temperatures. Therefore there is a thermodynamic driving force for Oi precipitation to occur. An unusual feature associated with oxygen precipitation, when compared to wellknown precipitation phenomena in metals or most dopants in semiconductors, is the presence of a large volume shortage. This results from the interstitial nature of Oi atoms in Si, which is not associated with a lattice volume. Thus only the Si atoms supply their atomic volumes to the formation of a Si02 precipitate. Since the molecular volume of Si02 is 2.2 times that of the Si atomic volume, there is a shortage of 1 Si atomic volume associated with the formation of each SiO, molecule. During the nucleation stage with a small number of Si02 molecules for each precipitate, this volume shortage is accommodated by elastic deformation of the Si matrix. Further growth of the precipitate will be prevented by the increase of elastic energy unless the elastic strain is relieved by plastic deformation, the emission or absorption of native point defects, and/or the incorporation of volume shrinking impurities such as carbon atoms (Fig. 5-36). In the following, we will deal with the two latter processes in a dislocation-free Si matrix, which may cause a supersaturation or undersaturation of the appropriate native point defects and impurities, which in turn may influence the nucleation and growth kinetics of precipitates. Let us first discuss the simple case that Si selfinterstitials relieve the

-

284


elastic stress completely. In terms of SiO, formation, this requires

2 0,+(1+/3) Si e.Si02+/31

(5-106)

where /3= 1.2. Assuming spherical SiOz nuclei and neglecting the influence of vacancies, we obtain the critical radius rent above which precipitates will grow as as2 Gnt

=

2 k , Tln[(Cl/C:q ) ( CFq/C,)''? ]

(5-107)

(GoseleandTan, 1982). In Eq. (5-107), (7 is the SiO,/Si interface energy for which 0.09-0.5 J m-* has been reported, and i'2 ( = 2 ~ 1 0 - m-3) , ~ is the volume of one Si atom in the silicon lattice. For the derivation of Eq. ( 5 - 102),p= 1 has been used. During precipitation a supersaturation of selfinterstitials will be produced ( C > CFq), which in turn will increase the critical radius for further nucleation. This may also cause shrinkage of already existing precipitates to occur if their radius is surpassed by the increased rcrlt(Ogino, 1982; Tan and Kung, 1986; Rogers et al., 1989). After a sufficiently long time and for a sufficiently high supersaturation, the self-interstitials will nucleate interstitial-type dislocation loops (usually containing a stacking fault, the bulk stacking faults), which will reduce the self-interstitial concentration back to its thermal equilibrium value. For C = Crq,Eq. (5-107) reduces to the classical expression for the critical radius. Vanhellemont and Claeys (1987) have given an expression for the critical radius in which besides selfinterstitials also vacancies and elastic stresses have been taken into account. A more detailed look at oxygen precipitates shows that their shape, ranging from rod-like, to plate-like, to being almost spherical, depends on the detailed precipitation conditions. The different shapes can be explained by a balance between minimizing the elastic energy and the point-defect

supersaturation (Tiller et al., 1986). The growth of oxygen pricipitates is limited by the diffusivity Di of oxygen interstitials given by Eq. (5-72). The growth kinetics of Si02 platelets has been measured by Wada et al. (1983) and Livingston et al. (1984), and theoretically analyzed by Hu (1986). Carbon precipitation is associated with a decrease of about one silicon atomic volume for each carbon atom incorporated in a S i c precipitate. The same volume decrease holds for carbon agglomerates without compound formation. The volume requirements during carbon precipitation, which are opposite to those during oxygen precipitation may be fulfilled by the absorption of one self-interstitial for each carbon incorporated (Fig. 5-36). When both carbon and self-interstitials are present in supersaturation, co-precipitation is a likely process to occur. B-swirl defects are thought to have formed in this way during Si crystal growth (Foll et al., 1981). If both carbon and oxygen are present simultaneously, it is obvious that co-precipitation of carbon and oxygen in the ratio 1 : 2 will avoid stress and point-defect generation or absorption. Co-precipitation of carbon and oxygen in this ratio has been observed by Zulehner (1 983), Hahn et al. (1988), and Shimura (1986). Hahn et al. (1988) also showed that the Si crystal remains essentially stress free in spite of a fairly large amount of co-precipitated carbon and oxygen. In a long time isochronal annealing experiment of CZ Si wafers containing a high supersaturation of both carbon and oxygen, intriguing features were observed (Shimura, 1986). Co-precipitation of oxygen and carbon, to a substantial amount and at the approximate ratio of 1 : 2, occurred at temperatures lower than -850°C. At still higher temperatures, however, only a significant precipitation of oxygen has occurred. Based on a method developed using the principle

5.10 References

of the maximization of the Gibbs free energy degradation rate (Huh et al., 1995a), Huh et al. (1 995 b) quantitatively explained the oxygen and carbon co-precipitation behavior, as observed by Shimura (1986), by considering the precipitate growth behavior. At and below 800 "C, dislocation formation is not possible and hence oxygen precipitates grow by the absorption of carbon atoms, together with the emission of some Si self-interstitials resulting in a Si self-interstitial supersaturation lower than that of the carbon-free Si case. Above 850"C, dislocations form during the initial time within which a sufficiently high Si self-interstitial supersaturation develops, and afterward this interstitial supersaturation diminishes via climb of the dislocations by absorbing the continually emitted Si self-interstitials caused by the Si02 precipitates, which continue to grow. Now, participation of carbon atoms in the Si02 precipitates is no longer needed, because the chemical energy of the precipitates will be higher in the presence of carbon atoms in the precipitates. In providing relief to the volume shortage associated with forming an Si02 molecule, the emission of a Si self-interstitial and the absorption of a carbon atom are two parallel chemical reaction type processes, and it is not a trivial matter to determine the relative contributions of the two processes in a selfconsistent manner. Up to now, it appears that such a determination can only be handled using the maximum Gibbs free energy degradation rate method, as was first attempted by Huh et al. (1995a, b).

-

-

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6 Dislocations Helmut Alexander I1. Physikalisches Institut der Universitat Koln. Koln. Federal Republic of Germany

Helmar Teichler Institut fur Materialphysik der Universitat Gottingen. Gottingen. Federal Republic of Germany 293 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.2 Experimental Results on the Electronic Properties of Dislocations 6.3 and Deformation-Induced Point Defects . . . . . . . . . . . . . . . . . 302 6.3.1 Electron Paramagnetic Resonance (EPR) Spectroscopy of Plastically Deformed Silicon . . . . . . . . . . . . . . . . . . . . . . . 308 6.3.2 Information on Dislocations and Point Defects from Electrical Measurements . . . . . . . . . . . . . . . . . . . . . . . . 313 6.3.3 Phenomena Indicating Shallow Dislocation-Related States . . . . . . . . . 322 6.3.3.1 Photoluminescence (PL) . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 324 6.3.3.2 Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 6.3.3.3 Microwave Conductivity (MWC) . . . . . . . . . . . . . . . . . . . . . . 6.3.3.4 Electric Dipole Spin Resonance (EDSR) . . . . . . . . . . . . . . . . . . . 326 6.3.3.5 Electron Beam Induced Current (EBIC) . . . . . . . . . . . . . . . . . . . 327 6.3.4 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 329 6.3.5 Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 A”BV’ Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Theoretical Investigations about Electronic Levels of Dislocations . . . 334 6.4 6.4.1 Core Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 334 336 6.4.2 Deep Electron Levels at Dislocations . . . . . . . . . . . . . . . . . . . . 6.4.3 Core Bond Reconstruction and Reconstruction Defects . . . . . . . . . . . 338 Kinks, Reconstruction Defects, Vacancies, and Impurities 6.4.4 340 in the Dislocation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Shallow Dislocation Levels . . . . . . . . . . . . . . . . . . . . . . . . . 344 6.4.6 Deep Dislocation Levels in Compounds . . . . . . . . . . . . . . . . . . . 344 6.5 Dislocation Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 346 6.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurements of the Velocity of Perfect Dislocations 6.5.2 in Elemental Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 347 6.5.3 Kink Formation and Kink Motion . . . . . . . . . . . . . . . . . . . . . . 350 6.5.4 Experiments on the Mobility of Partial Dislocations . . . . . . . . . . . . 352

292 6.5.5 6.6 6.6.1 6.6.2 6.7 6.7.1 6.7.2 6.7.3 6.7.4 6.8 6.9

6 Dislocations

357 Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Dislocation Motion . . . . . . . . . . . . . . . . . . . . . . . 358 Dislocation Motion in Undoped Material . . . . . . . . . . . . . . . . . . 358 Dislocation Motion in Doped Semiconductors . . . . . . . . . . . . . . . . 362 Dislocation Generation and Plastic Deformation . . . . . . . . . . . . . 365 Dislocation Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Dislocation Multiplication (Plastic Deformation) . . . . . . . . . . . . . . 368 370 Generation of Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . . Gettering with the Help of Dislocations . . . . . . . . . . . . . . . . . . . 371 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 371


List of Symbols and Abbreviations a

b, b Dk

Dn,Dnn d, d’ E,, E ,

E, G

K

Ed

EF Ef EO e ee* Fi

Fn

f J, k ki m N Nd NAs ND

Nc3 Nv

Q Qo

Q’ 4 4L Rc

Ri R* S* *n

*P

T Tm Tl tc

lattice parameter Burgers vector, Burgers modulus kink diffusivity dipole-dipole interaction between nearest- and next-nearest-neighbor vacancies along the dislocation line dissociation widths on cross slip plane and primary glide plane conduction and valence band edge critical activation energy formation energy for a critical double kink energy height of weak obstacles dangling bond energy Fermi energy formation energy dislocation level electron charge electron effective charge glide force normal component of Peach-Kohler force geometrical factor relating glide force difference and shear force nucleation rate and effective nucleation rate Boltzmann constant distribution coefficient stress exponent of dislocation velocity dislocation density density of chargeable sites acceptor density, donor density density of states at the conduction (valence) band edge activation energy of dislocation velocity activation energy of dislocation velocity at 1 MPa shear stress activation enthalpy charge line charge critical radius friction stress ratio of friction stress critical length capture cross section for electrons capture cross section for holes temperature absolute melting temperature spin-lattice relaxation time critical thickness

293

294

6 Dislocations

pulse length, pause length critical pulse length, critical pause length mean lifetime of a double kink activation energy of the lower yield stress intraatomic Coulomb integral formation energy of a single kink kink velocity, mean kink velocity activation enthalpy activation energy migration energy stacking fault energy lattice mismatch relative and vacuum permittivity strain rate plastic strain plastic strain rate shear modulus mobility Debye frequency kink attempt frequency applied compression stress excess stress shear stress threshold stress yield stress of CRSS band bending energy in the dislocation core Coulomb potential of a screened dislocation AD CL CRSS

cs cz

DK DLTS EBIC EDSR EPR FZ HFS HREM L LCAO LEC

antisite defect cathodo luminescence critical resolved shear stress constrictions Czochral ski double kink deep level transient spectroscopy electron-beam-induced current electric dipole spin resonance electron paramagnetic resonance floating zone grown hyperfine structure high-resolution electron microscope luminescence linear combination of atomic orbitals liquid encapsulation Czochralski


MWC PD PL PPE RD REDG SEM SF SI TSCAP V VVF

microwave conductivity point defect photoluminescence photoplastic effect reconstruction defect radiation-enhanced dislocation glide scanning electron microscope stacking fault self-interstitial thermally stimulated capacitance vacancy valence force field

295

296

6 Dislocations

6.1 Introduction At first glance the subject “dislocation in semiconductors” seems to be well defined. However, since dislocations in these materials are generated at elevated temperatures, their formation is always connected with a change in the number and distribution of point defects (intrinsic and extrinsic) in the crystal. Therefore, any change of a physical quantity measured at crystals with and without dislocations, respectively, has to be carefully analyzed as to which part of the change might be due to the dislocations themselves. The intricacy of this problem is illustrated by the current discussion of the extent to which clean dislocations are centers for carrier recombination. At present it is extremely difficult to arrive at a final answer to this question since methods for measuring the degree of decoration of dislocations with impurity atoms are lacking. In theory, the determination of the minimum energy core structure of straight partial dislocations has reached a state where fundamental changes are not to be expected. Regarding the electronic levels of straight dislocations, the theory indicates that no deep states exist (except perhaps for shallow levels due to extremely deformed bonds) for the so-called reconstructed variants of the dislocation cores, which seems to be the energetically stable configuration of the predominating partials in silicon. In the case of compounds, even more problems are unsolved than for the elemental semiconductors. Nevertheless, during the last few years considerable progress has been made in dislocation research. Extended application of EPR spectroscopy to plastically deformed silicon has led to a clear distinction between point defects in the bulk and defects introduced into the core of dislocations when they move. Moreover, a thermal procedure could be worked out which strongly reduc-

es the number of bulk defects after plastic deformation. As already mentioned, the core of straight partials in its ground state in silicon is reconstructed; after some motion it contains paramagnetic defects as (mainly) vacancies and other singularities as kinks and jogs. Theoretical calculations indicate that deep states in the gap arise mainly from broken bond centers. Accordingly, the intensity of the DLTS signals of deformed silicon is also greatly reduced by moderate annealing. Recently, shallow states in the gap connected to dislocations have come to the fore. Here, shallow is defined as being within 200 meV from the band edges. These states are responsible for near edge optical absorption, photoluminescence and transport phenomena such as microwave conductivity and combined resonance. Their nature is not yet clear and will be of central interest in the near future. Dislocations are extended defects, Le., their charge state may vary in wide limits. Consequently there is Coulomb interaction between charges confined to the dislocation which shifts the dislocation level with respect to the Fermi level (band bending around extended defects). The dependence of the EBIC contrast of dislocations on temperature and injection is analyzed along these lines. Careful analysis of the ionization of shallow donors (phosphorus) in plastically deformed n-type silicon by EPR clearly evidenced another source of band bending: The inhomogeneous distribution of deep acceptors (point defects) leads to inhomogeneous compensation of doping. Analysis of DLTS spectra under these conditions is not straightforward. For semiconductor technology the most interesting properties of dislocations are their activity as a recombination center and as a sink for self-interstitials. Research is active with respect to both these effects.

6.2 Geometry

In summary, it should be emphasized that in examining dislocations, the dislocation density does not satisfactorily characterize a crystal. Instead, its whole thermal history and the process which generated the dislocations have to be taken into account. Inhomogeneity after dislocation movement is unavoidable and may have a strong influence on physical properties. This message should be delivered before going into details. In this chapter we comment on both mechanical and electronic properties of dislocations, which are often dealt with in separate reviews. We felt that there is encouraging convergence of theory and experiment but that the field is far from being well-rounded. Thus separate presentation of experimental and theoretical results seemed most helpful for the reader to form an unbiased opinion.

6.2 Geometry The reader is assumed here to be familiar with the general conception of a dislocation (Friedel, 1964). As the most important semiconductors belong to the tetrahedrally coordinated crystals with diamond or sphalerite (cubic zinc blende) structure we have to deal primarily with dislocations in crystals with the face-centered-cubic (f.c.c.) Bravais lattice. As in f.c.c. metals, the Burgers vector of perfect dislocations is of the type a/2 (1 10). Also, the (close-packed) glide planes, determined by the Bravais lattice, correspond in diamond-like semiconductors as in f.c.c. metals to [ 11 1 } planes. However, the existence of two sublattices in the diamond as well as in the sphalerite structure brings about two types of glide planes. Figure 6- 1 shows a projection of the diamond structure onto a { llO} plane; obviously the (1 1) planes are arranged in

297

[ti11

Figure 6-1. Projection of the diamond structure onto a (1 10) plane.

pairs, the distance within one pair being three times shorter than between pairs. Thus at first glance the relative shift (shear) of one part of the crystal with respect to the other could proceed either in the wide space between two neighboring pairs of (171) planes or between the two planes of one pair. Since this shift is accomplished by the motion of dislocations those two possibilities result in dislocations with quite different core structures called “shuffle-set” and “glide-set” dislocations by Hirth and Lothe (1982). In one of the first papers considering plastic deformation of germanium Seitz (1952) came to the conclusion that dislocations in diamond-like crystals should belong to the shuffle-set (Fig. 6-2). For a long period of time that opinion was generally accepted because it was in agreement with two principles: First, the shear stress needed for displacement of two neighboring lattice planes, one against the other, is generally smaller, the wider the distance is between these planes; second, in the case of covalent bonding, it is reasonable to assume that cutting one bond (per unit cell of the plane) would be easier than cutting three bonds. In 1953 William Shockley gave a remarkable speech, documented as a short abstract (Shockley, 1953), in which he left open the question of whether dislocations in

298

6 Dislocations

dislocation moves one part of the crystal against the other by a Burgers vector which is not a translation vector of the space lattice; this partial dislocation (abbreviated as "partial") leaves the stacking sequence disturbed; it is followed by a stacking fault ribbon. This region of wrong stacking sequence is closed by a second partial, whose Burgers vector completes the first Burgers vector to a space lattice vector a/2 [Oll] Figure 6-2. 60" shuffle-set dislocation.

diamond-like crystals were of shuffle or glide type. He noticed that only in the second case is splitting of perfect dislocations into two (Shockley) partial dislocations possible, as in f.c.c. metals. This reaction, also called dissociation, consists of a decomposition of the elementary step of shearing into two steps: an initial partial

-+ a/6 [121] + a/6 [I121

(6-1)

Considering a model of the structure, it can easily be understood that only the two ( 1 i 1 ) planes constituting a narrow pair can be rebound by tetrahedral bonds after a relative displacement by a/6 [121]. This means that only glide-set dislocations are able to dissociate (Fig. 6-3). Thus the clear proof of the then new weak-beam technique of transmission electron microscopy that glissile dislocations in silicon on most of their length are dissociated initiated a fun-

*

4 0

Figure 6-3. Dissociated 60" glide-set dislocation. A stacking fault ribbon is bound by the two partials: on the left side the 30" partial, on the right side the 90" partial. (Partials are shown unreconstructed).

6.2 Geometry

damental change of dislocation models and theory (Ray and Cockayne, 1970, 1971). Speculation about the association of a shuffle-set dislocation with a stacking fault ribbon in a neighboring narrow pair of planes (Haasen and Seeger, 1958) made clear that the only difference between a dissociated glide-set dislocation and a shuffle-set dislocation which is associated with a stacking fault bound by a dipole of partials is one row of atoms (Fig. 6-4) (Alexander, 1974). EPR spectroscopy (Sec. 6.3.1) brings to light that when screw dislocations in silicon move, they introduce vacancies into the core of their partials. Thus an equilibrium between (dominant) glide segments and shuffle segments must be considered (Blanc, 1975). This equilibrium appears to be influenced by the tensor of stress acting on the dislocation as well as by the concentration of native point defects and possibly certain impurity species like carbon and oxygen (Kisielowski-Kemmerich, 1990). One may ask why the former decision in favor of the shuffle-set dislocation was a wrong decision. This becomes clear from Fig. 6.3. Admittedly, the number of “dangling” bonds is larger in this structure than in Fig. 6-2 by a factor of three. But the

Figure 6-4. Replacing the row of atoms in the core of a 30” partial by vacancies generates the core of a shuffle-set 60” dislocation associated to a stacking fault between a neighboring close pair of (1 1 1) planes (from Alexander, 1974).

299

arrangement of the orbitals containing the unpaired electrons is more suitable for pairwise rebonding (“reconstruction”) in the partials of glide-set dislocations because the orbitals show much more overlap than in the core of shuffle-set dislocations (Fig. 6-2). Reconstruction of the core of the basic types of partials has been studied theoretically, as described in Sec. 6.4.3. The idea that this reconstruction is realized at least in silicon is in satisfactory agreement with the results of EPR spectroscopy which always came out with a much smaller number of unpaired electron spins than geometrically possible dangling bonds (Sec. 6.3.1). In summary, one may state that dissociated glide-set dislocations are favored over shuffle-set dislocations because in glide-set dislocations, both elastic energy and energy of unsaturated bonds are saved. For germanium, however, the situation is not as clear. Proceeding from elements to compounds with sphalerite structure (A1”BV compounds and cubic A”BV1 compounds) an interesting complication arises: the two f.c.c. sublattices now are occupied by different atomic species. Nevertheless, in those compounds the overwhelming majority of mobile dislocations are also found dissociated into Shockley partials. Figure 6-3 can be used again, but the “black” and “white” atoms are now chemically different, and the bonding has an ionic contribution depending on the constituents. Obviously, all atoms which are seats of dangling bonds before reconstruction along a certain partial are of the same species. This makes reconstruction by pairwise rebonding more difficult; in fact, it is doubtful whether dislocations in compounds are reconstructed although without reconstruction, the preference for glide-set dislocations can no longer be easily understood. A further consequence of the uniformity of atoms occupying the core sites of a par-

300

6 Dislocations

tial is the doubling of the number of dislocation types when compared with elemental semiconductors: a 30" partial is conceivable with A atoms in its very core or with B atoms; the same holds for any dislocation. The naming of those chemical types is not uniform in the literature; there has been an attempt to remain independent from the decision whether the dislocations belong to the glide-set or to the shuffle-set. We take the view that dissociation into partials proves a dislocation to be of the glide-set type. Thus we call a dislocation with an extra half plane ending with A atoms (cations) pdislocation and its negative counterpart with anions in the center of the core a dislocation (Fig. 6-5). A more extended name would be A(g) and B (g), respectively. A B (s) dislocation would be of the same sign as A(g), but ending between widely spaced ( 11 1 } planes. a-60" dislocations dissociate into an 0-30" and an a-90" partial (Fig. 6-3), screw dislocations always consist of an a-30' and a p-30" partial. The cubic sphalerite structure is assumed (at room temperature) by several A'IB"' compounds as well: cubic ZnS, ZnSe, ZnTe, CdTe. But a second group (ZnO. hexagonal ZnS, CdS, CdSe) belongs to the (hexagonal)

A"

By

wurtzite structure. This structure also is composed of tetrahedral groups of atoms, but the stacking sequence of those tetrahedra is ABAB ... instead of ABCABC ... Here the basal plane (0001) it the only close-packed glide plane. It is equivalent to the four (1 11) planes in the cubic structures. The Burgers vectors of perfect dislocations are of the type b = ( d 3 ) (2110). Dislocations with these Burgers vectors may also glide on prismatic planes [ lOi0) and this secondary glide is indeed observed (Ossipyan et al., 1986). Since the ( l O T O } planes are chemically "mixed", there is no distinction between a and /3 dislocations on these planes. But there are two different distances between two (lOi0) planes in analogy to shuffle and glide planes in the case of (1 11). Information describing which one is activated i n wurtzite-type crystals is lacking (Ossipyan et al., 1986). Stacking faults (SF) generated by dissociation of glissile dislocations on close packed planes are of intrinsic type in all semiconductors investigated so far. An SF locally converts a thin layer from sphalerite into wurtzite structure and vice versa (Fig. 6-6). This is observed for ZnS, where one modification is transformed into the other by sweeping a partial over every second close-packed plane (Pirouz, 1989). Similarly, microtwinning of cubic crystals is equivalent to sweeping every plane by a Shockley partial (Pirouz, 1987). The stacking fault energy y in A'''BV compounds

c

1

c

B A 1

2

3

L

A

c

s

Figure 6-5. a and p (edge) dislocations in a cubic A'I'B" crystal. S: surfaces of the crystal. 1, 4: a dislocations. 2 , 3: Bdislocations. 1, 3: glide-set dislocations. 2,4: shuffle-set dislocations.

B A

c

B A

c

B

c

B A

-

A

c B

A

A

B

c

B

c

B A

A

c

B

c

B

A

c

B A

Figure 6-6. An intrinsic stacking fault generates a thin layer of hexagonal wurtzite (B C B C) in the cubic lattice.

6.2 Geometry

decreases systematically with increasing ionicity of the compound (Gottschalk et al., 1978). This becomes clear as soon as y is related to the area of a unit cell in the stacking fault plane and it can be understood from the local neighborhood of ions on both sides of the SF plane: here a 13th neighbor of opposite sign enters the shell of 12 next-nearest-neighbor ions, in this way reducing the Coulomb energy. Takeuchi et al. (1984; Takeuchi and Suzuki, 1999) extended this consideration to A"BV' compounds and showed good correlation of yto the charge redistribution coefficient s, wich accounts for the dependence of the effective ionic charge on the strain. Where a wurtzite phase exists the c/a ratio also is correlated to y. An interesting feature typical for dislocations in semiconductors are constrictions (CS) where the dissociation into partials locally is withdrawn (Fig. 6-7). CS can be point-like on weak-beam micrographs (Le., shorter than 1.5 nm) or segments on the dislocation line. It is well established that most CS are projections of jogs (Packeiser and Haasen, 1977; Tillmann, 1976). Consider-

301

ing density and distribution of CS in silicon after various deformation and annealing procedures we come to the conclusion that the majority of CS are products of climb events and not of dislocation cutting processes (Jebasinski, 1989). Point-like CS are jogs limiting a longer segment which redissociated after a climb on a new glide plane. Some of them may also be close pairs of jogs. Packeiser (1980) was able to measure the height of jogs in germanium and found them rather short (between 2 and 7 plane distances; elementary jogs were beyond the resolution of the technique). We recently measured the average distance L between two neighboring CS in p-type silicon and found that L = 0.6 pm for a strain of 1.6%, irrespective of whether the deformation was carried out at 650°C or at 800°C. Annealing after deformation leads to an increase of L for annealing below the deformation temperature, but to a decrease of L above Tdef, exactly as was found previously for Ge (Haasen, 1979). But only above Tdef (= 650°C) equilibrium was reached within the annealing time (16 h). It must be concluded that a fast process of annihilation of

Figure 6-7.Silicon. Transmission electron micrograph of a dipole of dissociated edge dislocations with constrictions. Left image: stacking fault contrast (g= (31)). Right image: weak beam contrast (022/033) (from Jebasinski, 1989).

302

6 Dislocations

CS (climb of jogs along dislocations) is superimposed to a slow process of generation of new CS (net climb of the dislocation). It is noteworthy that Farber and Gottschalk (1991) in CZ-Si observed only very few CS. There are extensive investigations of climbing of dissociated dislocations carried out with the help of high resolution electron microscopy (Thibault-Desseaux et al., 1989). The authors analyzed silicon bicrystals grown by the Czochralski technique and plastically deformed at 850°C. Climb proceeds by nucleation of a perfect interstitial loop on the 90" partial (Fig. 6-8). These loops may or may not dissociate. It is interesting that the climb events are found at dislocations which are trapped by formation of a dipole or near the grain boundary. Slowly moving or resting dislocations are preferentially concerned when interaction with point defects is considered, irrespective of whether this involves impurity atoms (Sumino, 1989) or native defects. ThibaultDesseaux et al. (1989) estimate the concentration of interstitials necessary for the first step of climb as 1O-'- IO-' and believe that such a supersaturation (lo6) is the consequence of plastic deformation. The above-rnentioned mechanism of climb via constrictions is not excluded by Thibault-Desseaux et al. (1989), but it is not particularly suited for investigation by HREM. It would be worthwhile to investigate whether climb in FZ-Si proceeds due to a supersaturation of vacancies, and in such a case CS would be prevalent. Dislocation core structures are studied by HREM i n end-on orientation, Le., the electron beam is parallel to the dislocation line (see Spence, 1988). The micrograph shows a projection of the atom colums (Fig. 6-8). Jogs as well as kinks cannot be resolved in this orientation. Recently, dissociated dislocations have been imaged i n plane-view

Figure 6-8. Silicon: Climb by formation of a complete dislocation (high-resolution TEM). (a) dissociated 60" dislocation (Ah: 90" partial, 6B: 30" partial); (b) The partial A 6 has decomposed into a complete dislocation AC (which has climbed by 7 atomic planes) and the partial C6 (from Thibault-Desseaux et al., 1989).

orientation with high resolution (Alexander et al., 1986; Kolar et al., 1996; Spence et al., 1997). Here the electron beam is perpendicular to the glide plane, the stacking fault ribbon between the partials shows up as an area with hexagonal arrangement of the atom columns. The boundaries of that area marks the core of the partials (for application see Sec. 6.5.4).

6.3 Experimental Results on the Electronic Properties of Dislocations and DeformationInduced Point Defects Dislocations in semiconductors act as electrically active defects: they can be

6.3 Experimental Results on the Electronic Properties of Dislocations

“structural dopants” (acceptors and/or donors), recombination centers reducing the lifetime of minority carriers, or scattering centers. In the low-temperature region, dislocations are linear conductors. In addition to this direct influence on carrier density, lifetime, and mobility, there are indirect influences: electrically charged dislocations are surrounded by a screening space charge which causes local band bending and therefore may change the charge state of point defects in this region. This multitude of electrical effects has attracted considerable research activity for a long time; however, on account of some special problems, understanding has developed rather slowly. The first problem, which had not been realized immediately, concerns the superposition of the electrical effects of point defects also produced by plastic deformation with that of dislocations (plastic deformation is the usual method to produce a sufficiently high density of welldefined dislocations). As revealed mainly by EPR spectroscopy, but also by Hall effect measurements, deformation produces a surprisingly high concentration of point defect clusters contributing to and sometimes dominating the electrical properties of deformed crystals. The second problem arises from the character of dislocations as extended defects: While a point defect may change its charge state by one or two elementary charges, dislocations act as traps for majority carriers along their line; i.e., they may concentrate a considerable charge along their line. In this section we will first give a short summary of the most elaborate theories on the effects of an electrically charged dislocation on the distribution of the electron states in the band scheme. Plastic deformation of semiconductor crystals introduces deep states into the gap. It has been a major research goal for

303

decades to establish whether these states are due to dislocations and, if the answer is yes, whether they are due to the dislocations themselves or due to their decoration by point defects including impurity atoms. Experiments are aimed at measurements of the carrier density (Hall effect), and the effects of the states as traps (DLTS) and as recombination centers (EBIC). Interpretation of the experimental results requires comparison with theoretical concepts concerning the electrical properties expected for a linear arrangement of deep states. In the following we give a survey of some important models proposed so far. The pioneering work in this field was published in 1954 by Read. At that time, the structure proposed for the dislocation core in diamond-like crystals was as shown in Fig. 6-2. The distance c between neighboring dangling bonds varies between 0.4 nm and infinity depending on the dislocation character. Read ascribed to those sites with unpaired electrons a single acceptor level, i.e., they are neutral when occupied by one electron and negative with two electrons. Dislocations, therefore, are lines of variable negative charge (model I). Since the distance c between chargeable sites is much smaller than the distance between pointlike acceptors in the bulk, the electrostatic interaction between charged sites prevents degrees of high occupation: in practice the ratio between c and the distance a of neighboring charges is of the orderf= cla I0.15. Lattice defects with the peculiar property that the occupation of their chargeable sites is limited by charge interaction are called extended defects (ED). Read calculates the actual occupation f of the dislocation for a given doping (IND-NAI) and temperature from the energy minimum of the system. This approach is exact only at T = 0; at a higher temperature a certain probability for electron hopping along the dislocation must

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6 Dislocations

be taken into account, and entropy will lead to a minimum of free enthalpy at a nonequidistant arrangement of the charges. Starting from the same structure of the dislocation core Schroter and Labusch (1 969) came to a different theoretical approach (model 11). As proposed by Shockley (1 953), these authors took the periodical arrangement of equal electron states seriously and proposed a one-dimensional band which is half full (f’=1/2) when the dislocation is electrically neutral (f=0 in model I). This band can give up ( 0 5 f’e 1/2) or accept electrons (1/2 0, EF(T) = E,-kT In (C, T3”l(n)) for n-type, where ( ) means averaging over large volumes.] The potential qoof an electron within the chain of electrons along the dislocation can be determined by adding the interaction of the electron with the other electrons and with the positive screening charge (Read, 1954)

The next step in constructing a model of the charged dislocation concerns the role of (configurational) entropy as soon as the temperature is clearly above zero (as in experiments). Read (1954) states that at T > 0 the nonequidistant arrangement of electrons will bring an increase of the interaction energy as well the appearance of an entropy term AS. Both contributions to free enthalpy at least partly cancel, so that the minimum energy approximation (MEA) (referred to earlier) should be adequate as long as kT is small compared tofE,. Read then deduces a different model which underestimates the free energy - in contrast to MEA. We have no space to follow the discussion further (Read, 1954; Labusch and Schroter, 1980; Figielski, 1990). At a low occupation ratio f, the entropy plays an important role if a full band is considered. In conclusion we will give a quantitative example: Consider n-type silicon ( E = 12) with an effective doping IN,-N,I = 10l6 cmP3.Read’s “standard” (60”)dislocation is characterized by 0.385 nm between dangling bonds. We assume the dislocation

level ET at 0.23 eV below the edge of the conduction band E,, and the Fermi level EF (at 0 K ) at 0.03 eV below E,. Consequently, the dislocation level is shifted in equilibrium by 200 meV upwards. From Eq. (6-9) we calculate the (equilibrium) occupation ratio,f=O. 107. In other words, the distance between two neighboring excess electrons a = 3.6 nm, and the dislocation line charge q L= 4.44 x lo-’ A s m-I. The Read radius R turns out to be 94 nm. The electrostatic energy of the screened dislocation E,, = 80 meV per excess electron. As shown in Fig. 6-9 the shift of the dislocation level with respect to the undisturbed valence band edge is the sum of E,, and the bending of the band edges (-eqB) at the dislocation line. From this we calculate -eqB=(200-80)meV= 120meV. This value is confirmed by numerically computing (Langenkamp, 1995) the potential qB in the center between two charges within a chain of equidistant point charges: ?#B=-el(4n~o& a )[ln (Rla)+0.272]. The energy e qBindicates the energy barrier that electrons have to overcome on their way to the dislocation in the conduction band. However, the potential qo of an excess electron in the dislocation is -176 mV, this value is smaller than the total energy shift of the dislocation level (200 mV). Read explains the latter difference by a gedanken experiment: Take one electron from a charged dislocation without moving the other electrons. Since the energy of an electron in the dislocation is higher than at r +00, the system looses the energy e qo.If we add the rearrangement of the remaining electrons to equal distances, the energy of the system decreases further. The work released by the combined action is d/df(fEel)=EF-ET, i.e., the shift of the dislocation level. Therefore, the latter quantity must exceed e Yo.


307

E

Figure 6-9. Band bending at a charged dislocation. e ly,: band bending and energy barrier; Eel: electrostatic energy of the system consisting of the charged dislocation plus screening charge; EF:Fermi level (after Langenkamp, 1995).

For comparison with experiments, the activation energy AE for lifting an electron from the charged dislocation into the conduction band will be important, and is given by the energetical distance of the conduction band edge E, from the dislocation level ET, both taken at the dislocation. As demonstrated by Fig. 6-9 (6-1 1) (Ec-ET)la = AE ( u ) = A E (U + -) - E,,(u) This means that the activation energy depends on the charge of the defect. This statement is equivalent to the following: It is impossible to think of the dislocation level ET and the local band edges as moving rigidly together (“rigid band model”). Touching on experimental examination of the various models proposed so far, we have to first mention extended measurements of the carrier density of deformed germanium and silicon, both n-type and p-type, as a function of temperature using the Hall effect (Schroter, 1967; Schroter and

Labusch, 1969; Weberet al., 1968; Labusch and Schettler, 1972; Labusch and Schroter, 1980). From the Hall effect, the carrier density n ( T ) or p ( T ) can be determined. Comparing crystals before and after deformation, the change of the carrier density can be measured, which is assumed here to be due to dislocations. If the dislocation density N is known, the average occupation ratio f can be calculated. [Problems that arise due to the influence of point defects and impurities, as well as concerning measurements of N , are discussed by Labusch and Schroter (1980).] The strongest argument in favor of model I1 (half-full band at the neutral dislocation) is delivered by measurements of p-type material. In model I, the dislocation can only accept electrons, whereas model I1 looks at the dislocation as an amphoteric center: fcan be both positive and negative. Because of this difference, main efforts has been directed at p-type germanium. Lowdoped material with a relatively small dis-

308

6 Dislocations

location density is used. Qualitative observation shows the hole density p of deformed crystals to decrease over the exhaustion range (p = consr) of the undeformed crystal. 0 Pdef>Pundef; At high temperature, ( ~ 2 0 K) p d e f decreases to a low temperature and crosses the value P u n d e f at a certain temperature To. Obviously here the dislocations are neutral (f=0). The position of the Fermi level at To coincides with the dislocation level ET of the neutral dislocation. For germanium, E T was found 0.09 eV above the valence band. Further details can be found in Labusch and Schroter (1980). In summary, model I1 was proven to be much better at describing experimental results than model I. In spite of this success, it must be held in mind that the structural model of the dislocation core taken as the basis of all theories described above has several weaknesses. Some of them are trivial, some came to light after the “golden age” of these theories. It is well known that there is no way to introduce dislocations all belonging to the same type into a crystal. So a whole spectrum of densities of chargeable sites has to be expected. More important is dislocation splitting: TEM shows that practically all dislocations in (elemental) semiconductors are dissociated into two parallel partial dislocations with a stacking fault* in between. Most importantly: Theory and experiment suggest that most of the topological dangling bonds are reconstructed forming covalent bonds without deep states in the gap. It is highly probable that dislocations (additionally to shallow bands) contain deep states, but these are not periodically arranged along the (partial) dislocations. Rather they belong to localized secondary

* It is possible that the stacking fault also carries states in the gap.

defects (reconstruction defects, kinks, vacancies, etc.), of which apparently many more types are conceivable than hitherto known (Bulatov et al., 1995). Another effect which would obscure results is the inhomogeneity of effective doping arising from the inhomogeneous production of (electrically active) point defects by plastic deformation (Kisielowski et al., 1991). So future research on the theoretical side will be directed to ab initio calculations of defect models revealing the presence and position of deep states (Csanyi et al., 1998), and on the experimental side to local tests of single dislocations, e.g., by EBIC (Kittler and Seifert, 1993 b). The next section will show what information on the existence of unpaired electrons (and their surroundings) comes from electromagnetic paramagnetic spin resonance (EPR). After that, in Sec. 6.3.2 we will go on to discuss some of the modern electrical measurements of plastically deformed semiconductors (DLTS, EBIC).

6.3.1 Electron Paramagnetic Resonance (EPR) Spectroscopy of Plastically Deformed Silicon Where EPR spectroscopy is applicable, it yields more information on the defect under investigation than any other experimental method, because it reveals the symmetry of the defect. Via hyperfine structure (HFS) of the spectrum it also provides a hint as to the chemical species of atoms involved. Moreover, EPR spectroscopy can be calibrated to give numbers of defects. Admittedly, EPR concerns only paramagnetic centers; this means that there may be electrically active defects not detectable by EPR. Other defects will be traced only in a certain charge state. The charge state may reveal something about the position of the defect in the energy gap by observing EPR


spectra under illumination and of doped crystals . Fortunately, silicon is one of the most suitable substances for EPR. Thus we will begin with a summary of what is known about the EPR of plastically deformed silicon crystals (Kisielowski-Kemmerich and Alexander, 1988). So-called standard deformation by single slip ( T = 650°C, t = 30 MPa, AUl= 5 % ) resulting in a dislocation density N = 3 x lo9 cm-2 produces about 10l6cm-3 paramagnetic centers. The majority (65-80%) are point defect clusters of high thermal stability. The remaining defects are related to the dislocation geometry by their anisotropy. A clear distinction between the two classes of defects can be made by several methods: (a) As mentioned before, the dislocation-related defects show anisotropy with respect to the (total) Burgers vector of

309

the primary dislocations as a prominent axis. (b) Taking advantage of the fact that the spin-lattice relaxation time T , of these defects is up to four orders of magnitude shorter than that of the point defect clusters, Kisielowski-Kemmerich succeeded in separating the two parts of the spectrum completely (Fig. 6-10). On the other hand, using special passage conditions, one can detect both parts of the EPR spectrum at 15 K (they are normally seen at room temperature and at helium temperature, respectively). This disproves the assumption that the transition from the high-temperature spectrum to the low-temperature spectrum was due to a magnetic phase transition of the magnetic moments along the dislocation lines. (c) A two-step deformation interrupted by an annealing treatment suppresses to a large extent the production of the point defect spectrum during the second deformation

Figure 6-10. Silicon. EPR. (a) Dispersion spectrum at different temperatures. (b) Anisotropy of the spectrum of point defects (long-spin lattice relaxation time) and of the dislocation-related centers (short T I )(from Kisielowski-Kemmerich et al., 1985).

31 0

6 Dislocations

step without changing the dislocation spectrum (Kisielowski-Kemmerich et al., 1986). (d) The attribution of the room temperature spectrum to point defect clusters is made more convincing by the production of a point defect cluster which is already well known from neutron irradiated Si. This, in fact, is possible by deformation at 390°C (Brohl et al., 1987). (Radiation defects anneal out around 400°C.) (e) Finally, the possibility to recharge the defects by doping is different for the two types of the paramagnetic defects. EPR spectroscopy was applied to deformed silicon mainly by two groups for over more than two decades. Most of the experimental results were i n perfect agreement. But in contrast to our interpretation given above and based on the anisotropy of the spectra and on variation of the deformation procedure, the Chernogolovka group insisted on ascribing the EPR spectra taken at room temperature and at low temperature to one and the same group of paramagnetic defects, located linearly along the dislocation cores (Ossypian. 1982). If all EPR centers are added, their number in fact corresponds to the number of dislocation sites. The change in the spectrum at around 60 K was ascribed to a magnetic phase transition which reduces the number of unpaired electrons because of the reconstruction of most of the dangling bonds. Admittedly, each EPR investigation on dislocations started with such a model in mind, but we feel the experimental results described above provide a convincing reason to abandon the model. The most important conclusion to be drawn from the EPR spectrum is the following: most of the geometrically possible broken bonds in the core of partial dislocations are reconstructed or are at least lacking unpaired electrons. While most authors take that as proof for pairwise rebonding

along the cores of partials, Pohoryles ( 1989) concluded from measuring photoconductivity of deformed germanium and silicon in helium gas under pressure that only helium atoms drive the reconstruction of the otherwise unreconstructed cores. He discusses negative U behavior as a reason for the EPR results. According to our interpretation, about 3% of the core sites of dislocations are occupied by unpaired electrons. It is worth noting here that the parameters describing the dislocation-related EPR spectrum are perfectly reproducible and do not depend on doping or on variation of the deformation conditions (the only exception being the number of centers). This is remarkable since plastic deformation causes significant strain in the lattice. Evaluating now the anisotropy of the low-temperature (dislocation related) spectrum we can identify three different contributions: first, a wide line (10 G) similar to a certain extent to the EPR signal of amorphous silicon is apparent; the related center was called Si-Y by Suezawa et al. (1981). On top of this wide line, several narrow lines ( 1 G) stand out (Si-K l), two for each activated slip system. Finally, a series of pairs of lines mark paramagnetic centers with spin S 2 1 (Si-K2). These lines also have a width of 10 G. It is possible to transform by light Si-K 1 centers into K 2 centers (Erdmann and Alexander, 1979). Two pieces of information are most important for modeling the paramagnetic defects: The g tensor of Si-Y and Si-K2 is of orthorhombic I (C2J symmetry, with the axis where g is nearest to the free electron value g, along [Oi 11, therefore perpendicular to the (total) Burgers vector [Ol 13 of the primary dislocations. Hyperfine structure identifies Si-K1 as a center of dangling bond type, the orbital being 22" from a (1 l l } bond axis. (The two lines belonging


to the primary dislocations are due to centers pointing "parallel" to [ l l i ] and [ l i l ] . These two directions are perpendicular to b as well (Fig. 6-1 1); Weber and Alexander, 1979). The line pairs Si-K 2 are most closely related to the Burgers vector: the axis of the fine structure tensor is exactly parallel to the total Burgers vector b (Bartelsen, 1977). It is important to realize that the total Burgers vector of the dissociated dislocations does not influence the atomic neighborhood of any atom except constrictions (cf. Sec. 6.2) which, however, cannot be identified with the EPR centers because of their number. Rather, the atomic structure in the core of a partial dislocation is determined by the partial Burgers vector, being of the type (21 l}. Thus we came to the conclusion that the distinction of the total Burgers vector in the spectra must mean that the related paramagnetic centers are located in the screw dislocations which run parallel to the Burgers vector (Weber and Alexander, 1979). Considering the core of 30" partials (Fig. 6-3) constituting screw dislocations, one notices that the broken bond of a reconstruction defect points along [Oll]. The dangling-bond-like orbitals Si-K 1 are nearly perpendicular to this direction. This observation suggested a vacancy in the core of such a 30" partial as the defect producing the Si-K 1 signal. Kisielowski-Kemmerich (1989, 1990) started on this basis a group-theoretical analysis of the defect molecule consisting of a nearly planar group of 4 atoms corresponding to the arrangement of the innermost atoms in the core of a 30" partial (Fig. 6-1 1). In a first approximation, the defect has a threefold rotation axis imbedded into the crystal parallel to a twofold crystal axis. This situation is abnormal in solid state physics but occurs here because (on account of the stacking fault) one of

A

t

[lit,

31 1

C

B Figure 6-11. Model of the core of a 30" partial tackled by Kisielowski-Kemmerich (1990). Atom C belongs to the stacking fault. (All angles 120°.)

the atoms (C in Fig. 6-1 1) is not in a regular lattice position. First, the reconstruction defect (7 valence electrons in the defect molecule) was considered; in agreement with the qualitative argument above, it turned out that the g tensor (reflecting the local crystal field) must have its g, axis perpendicular to the plane of the molecule. The author then investigated the defect molecule of Fig. 6-1 1 removing the central atom D. In other words the complex defect consisting of a reconstruction defect (a soliton) and a vacancy (Le., a three-fold coordinated vacancy in the core of a 30" partial) is under consideration. The defect molecule is planar suggesting sp2 hybridisation. It contains three valence electrons. This situation is unstable against Jahn-Teller distortion. Discussing several possibilities for that distortion Kisielowski-Kemmerich comes to the conclusion that extension of the distance between the atoms A and B compared to the other distances results in the right symmetry of the unpaired electron corresponding to the EPR line Si-Y. From the viewpoint of solid-state physics, the most-interesting EPR center in deformed Si is Si-K 2 because coupled spins are involved. We can either treat the spec-

312

6 Dislocations

trum as consisting of S = 1 line pairs ascribing the difference between the six pairs to different surroundings for the different pairs, or we can assume the coexistence of groups of coupled spins from n = 2 up to n = 6 (Bartelsen, 1977) (only one group of Si-K 2 is considered here with its maximum at 20 K). The latter model is accepted by Kisielowski-Kemmerich (1989), and it is interpreted as n threefold coordinated vacancies (Y centers) aligned along a dislocation core. The symmetry remains the same as for one Si-Y center (orthorhombic I = C2,,). The interaction between the electrons in their orbitals parallel to [Oil] is of pure x character and is weak. The total spin is S = n 1/2. Compared with vacancy chains in irradiated silicon investigated by Lee and Corbett ( 1976), one should bear in mind that in that case the vacancies are arranged along zig-zag chains with average [Ol 11 direction, but in our case two vacancies are occupying next-nearest sites along a [Oll] line. Taking this into account, the interaction parameter D, and D,, as determined by Bartelsen (1977) show the right ratio". Their absolute magnitude is smaller by about a factor of two, which may be due to a different form of the wave function of the unpaired electrons. Two recent papers question this interpretation of the EPR signal Si-Y and the line group Si-K 2 correlated with it. Lehto and Oberg ( 1997), on the one hand, calculate the structure of a vacancy in the core of a 30" partial using hydrogen-terminated clusters and local density-functional theory. The distortion of the triangle ABC (Fig. 6-1 1 ) resulting from this calculation is different from that deduced by Kisielowski-Kemmerich from the EPR signal: the distance AC

* D, and D,, measure the (dipole-dipole) interaction between nearest- and next-nearest-neighbor vacancies along the dislocation line.

is 46% longer than ideal bond lengths, and the other two distances are 28% longer. Most importantly, the unpaired electron is in the "bond" AC, as revealed by supercell calculation. This means that the three-fold coordinated vacancy in the core of a 30" partial should be an EPRcenter, but with symmetry different from Si-Y. The second important paper is by Cshnyi et a]. (1998). These authors propose a new structure of the reconstruction defect (RD, soliton) of the reconstructed 30" partial in silicon. The authors carried out ab initio calculations applied to supercells. The surprising result in the context of EPR is the following: The central atom D (Fig. 6-1 l), the so-called soliton atom, moves out of the ideal dislocation line somewhat approaching a neighboring atom belonging to the stacking fault, but different from atom C of Fig. 6- 11. This atom is now formally fivefold coordinated. The unpaired electron is shared between this atom and the soliton atom. Its character is primarily s. Therefore, the related EPR signal must be of low anisotropy. The authors recall the presence of fivefold coordinated atoms in amorphous silicon. Now, it is well known that the plastic deformation of silicon above 800"C, as well as the annealing of crystals deformed at a lower temperature, exclusively produces an EPR signal that is practically identical to the EPR line of a-Si (g = 2.005, low anisotropy, line width 7-10 G) (Alexander, 1986; Kisielowsky-Kemmerich and Alexander, 1988). [This signal has been named Si-R and Si-0 by Osipyan (1982).] Moreover, the signal Si-R appears to be a residual of the signal Si-Y, as found after standard deformation. (The line width of Si-Y strongly increases above 180 K, also in accordance with the line of a-Si.) From these facts, Csinyi et al. (1998) deduced their proposal that the strong signal Si-Y is


due to solitons in the core of reconstructed 30" partials. After standard deformation, the mean distance between two Si-Y centers is 40 lattice sites averaging over the total length of partials. After deformation at high stress (300 MPa) and 420"C, the dislocations are straight at a mesoscopic level and contain an order of magnitude less Si-Y centers (solitons?) (Weber and Alexander, 1983). Unfortunately, it is not possible to determine the exact fraction of 30" partials. If the identification of Si-Y by Csinyi et al. (1998) is correct, the interpretation of the other dislocation related EPR centers (Si-K 1 and K 2) is open anew. The numerical ratio of the two appears to depend on the charge, as shown by the reversible transformation by light with a characteristic energy of 0.65 eV (Erdmann and Alexander, 1979). We must stress that the paramagnetic centers are not intrinsic ingredients of dislocations; rather, they are produced during dislocation motion. Comparing different deformations under equal stress shows that the density of magnetic defects is proportional to the plastic strain, Le., the area of glide planes that the dislocations have swept. Alternatively, comparing deformations under different stress shows that the production rate increases with stress. The centers can be annealed - with the exception of a small part of Si-Y - at or above 750"C, A second deformation after this annealing starts with the high dislocation density reached at the end of the first deformation, but the density of magnetic defects now starts at zero and increases with as during the first deformation (KisielowskiKemmerich et al., 1985). Unfortunately, up to now EPR spectroscopy has yielded information on dislocations only in silicon. For germanium, the high nuclear spin of the isotope Ge73and the strong spin-orbit interaction are impeding factors, Similar problems arise with III-V-

31 3

compounds. Moreover, no method is known here to eliminate the spectra of point defects, which are more numerous than in Si.

6.3.2 Information on Dislocations and Point Defects from Electrical Measurements Since the early days of semiconductor physics an impressive number of careful investigations have been devoted to the question of which defect levels in the energy gap are due to dislocations. Because dislocations interrupt the translational symmetry of the crystal, such levels are to be expected, and in 1953 Shockley proposed the broken bonds in the core of dislocations to act as acceptors. In fact, measurements of the Hall effect seemed to confirm this idea. Schroter (1 967), investigating the Hall effect of deformed p-type germanium crystals in a wide temperature range, concluded that the dislocation states in the gap form a onedimensional band which can compensate for shallow acceptors at low temperatures, and which can also accept electrons from the valence band at higher temperatures. The occupation limit of this amphoteric band when neutral was found to be 90 meV above of the valence band edge. Ono and Sumino (1980, 1983) tried to evaluate along the same lines Hall data of p-type silicon crystals deformed at 750°C. They concluded that this was impossible, because they were not able to fit the temperature dependence of the density of free holes. Instead, experimental results and theory could be brought into reasonable agreement under the assumption that plastic deformation produced point-like electrical centers. This means that no shift of the level of those defects in the gap by Coulomb interaction should be inferred, which is typical for extended defects. Quantitatively, it turned out that simultaneously with

31 4

6 Dislocations

5 x lo7 cm-* dislocations (only etch pit densities have been determined) about 5 x l O I 4 cm-3 acceptors and 7 x 1013cm-3 donors are produced. The energy levels of both types of point-like centers are approximately at the same position (0.3-0.4 eV above the valence band edge). The decision in favor of point-like centers is natural because in view of the low ), a doping of the material ( I O i 4 ~ m - ~such large density of acceptors in the dislocation lines would cause a band bending far exceeding the width of the band gap. Thus, it can be stated that two methods as different as EPR spectroscopy and Hall effect measurement lead to the same important conclusion: at least in silicon, the consequences of plastic deformation for the electrical properties of a crystal are mainly due to point defects (PD) not located along dislocation lines. That does not exclude that other electron states do exist which are localized along dislocations (as are the EPR centers Y, K 1, and K 2). As shown by Wilshaw and Booker (1985), the temperature dependence of the EBIC (electron-beaminduced current) contrast of dislocations in deformed silicon can be accounted for by a certain number of rechargeable centers along the dislocations. These centers are subject to a shift in the gap equivalent to band bending by Coulomb interaction of the charges on the dislocation. (Obviously a charged dislocation assumes the character of a continuously charged line only if either the wave functions of the dislocation states overlap sufficiently to delocalize electrons along the line or if the actually charged (localized) states are less distant than the Debye screening length.) For completeness, a third class of electronic states in the gap after plastic deformation should be mentioned: these states must be close to the edges of the valence band and the conduction band, and they

form one-dimensional bands. Information on these shallow bands comes from microwave conductivity, electric dipole spin resonance, photoluminescence and optical absorption (cf. Sec. 6.3.3). Because most of these effects exhibit strong correlation with the dislocation geometry, these bands must be in the proximity of dislocation lines. For silicon, we are now in a position to indicate a procedure which avoids those difficulties to a large extent: Deformation at 800°C and annealing at the same temperature for 16 h greatly reduces the number of stable point defects. If subsequent deformation at lower temperatures is restricted to small strains, the number of deep point-like traps is small (Kisielowski-Kemmerich et al., 1986). As soon as band bending is assumed to exist, another problem should be considered: If the dislocation essentially contains several electrical levels, filling of the lowest level by band bending may prevent higher levels from being filled. On the basis of this model, Wilshaw and Fell ( 1989) could exlain that the EBIC contrast of (internal) dislocation loops in n-type Si decreases with intensification of the electron beam which induces electron-hole pairs: the dislocation charge is reduced below the equilibrium value (EBIC contrast was shown to be proportional to the line charge qL). With temperature, the EBIC contrast changes on account of changing qL, too. From those experiments and the model, the authors deduce a density Nd = 5 x lo7 m-I for the dislocation loops in Si. The level E, can only be limited to be deeper than 0.3 eV. For a (local) dislocation density of N = 5 x 1O6 cm-2 the volume density of dislocation centers turns out to be 2 x 10l2 ~ m - ~ . Such a density of recombination centers would never be detectable if it were not due to band bending. It has been questioned whether clean dislocations (i.e., free of

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impurity atoms) can cause an EBIC contrast at all (Kittlerand Seifert, 1981). Such acontrast arises when the specimen contains recombination centers, whose effective capture cross section is to be compared with the square of the diffusion length of minorities. Assuming the cross section of the order of the core atom of a dislocation, the contrast would be undetectable with present means. However, as can be understood from the foregoing, a charged dislocation acts through the electric field in its screening space charge, the cross section of which is lo5 to lo6 times larger than an atom. Thus a continued effort should be made to distinguish EBIC contrasts from decorated dislocations and from clean dislocations. A direct demonstration of the electrical field in the depletion region around highly dislocated layers within silicon crystals is given by the bright EBIC contrast of those regions outside the Schottky contact. At the same time this phenomenon shows conductivity along such layers (Alexander et al., 1990). Recently DLTS (deep level transient spectroscopy) has become the most effective technique to establish the density and position of levels in the gap. Its advantage before EPR is that the charge state of the defect to be investigated is not restricting. Defects, which are always sensitive to any annealing, are investigated using Schottky diodes on the crystal surface. In n-type material, electron traps in the upper half of the gap are detected, and in p-type crystals, the lower half of the gap is traced. Unfortunately, there are severe problems when applying DLTS to deformed specimens: (1) As we have seen, plastic deformation by some percentage produces lOI5 to 1016 cm-3 deep traps (mainly point defect clusters). Doping should clearly exceed the number of traps, otherwise compensation must be taken into account. Therefore,

31 5

weak deformation is optimal for DLTS; but then comparison with the less-sensitive EPR becomes difficult. (2) Because of band bending, the position of levels near to extended defects will change during filling and emptying; moreover, this effect can be frequency dependent. Band bending by a small number of deep levels can obscure many shallow levels (Shikin and Shikina, 1988). In any case, superposition of band bending by the defect and by the depletion region of the Schottky barrier must be analyzed (Nitecki and Pohoryles, 1985). (3) As shown first by Figielski (1978) and since then confirmed by experiment, the filling characteristics of extended defects are logarithmic in time. This makes calibration of the number of traps often approximate. (4)Finally, impurities can be confusing (Kronewitz and Schroter, 1987). ( 5 ) As outlined below for EDs the convenient standard evaluation of the spectra (Arrhenius plot) is not applicable. Hedemann and Schroter (1997) promise to give methods to determine defect parameters from fitting of experimental DLTS-data of EDs, both bandlike and localized. Before presenting the experimental results obtained by DLTS of plastically deformed silicon and GaAs, we will refer to recent theoretical approaches to the evaluation of DLTS spectra of extended defects. An important classification of EDs concerns the population dynamics of the electron states (Schroter et al., 1995). In case of EDs, there is not one sharp energy level, as in case of isolated point defects, but a distribution N D ( E )of states. The distribution can be either bandlike if the wave functions are extended in the core of the defect, or it consists of localized states, e.g., due to imperfections in the core or due to point defects interacting with the ED. The two

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31 6

6 Dislocations

classes of states can be distinguished by comparing the time & of internal equilibration of occupation with the inverse rates of exchange with the related band ( R e is the emission rate and R, is the capture rate). If I-, is shorter than R-' the states are bandlike, if on the other hand exchange is the quicker process, then the states are isolated. Because the characteristic times defining the difference cannot be measured directly, simulated DLTS spectra must be compared with assumed parameters. From this, some empirical rules were obtained. Varying the length tp of the filling pulse influences the high-temperature side of the DLTS line in a characteristic manner: for bandlike states, this side of the line is mainly independent of rp (Hedemann and Schroter, 1997), while for localized states the high-temperature side of the line coincides for different tp values after normalization with respect to the line maximum (Schroter et al., 1995; Hedemann and Schroter. 1997). Applying conventional analysis (Arrhenius plot of the emission rate e , over T 2 against T - I ) yields results that strongly depend on tp. This standard analysis therefore gives for both classes of ED, no more than a kind of finger-print if the applied pulse length is specified (Hedemann and Schroter, 1997). If an extended defect consists of localized states, those states are only coupled via the common energy barrier; therefore, it is necessary to include this barrier in the rate equation used for simulation of the DLTS signal (Hedemann and Schroter, 1997). Simulation* of capture and emission processes needs an idea about the density of states D ( E ) of the defect. The most simple assumption is a box-like function N , ( € ) . A * The term simulation should be reserved to solving the rate equation of capture-emission processes (e.g., Schroter et al., 1989). It should be distinguished from "fitting" a line to, e.g., a Gaussian shape.

more physical model has been developed by Kronewitz (1991) and Schroteret al. (1995) who assume that point defects distributed over the dilatation field of edge dislocations in silicon produce the C-line in the DLTS spectrum. The authors deduce N , ( E ) as proportional to ( A E i ) - 3 , where A E i means the energy shift of the i level from its situation far from the dislocation. This example illustrates the idea underlying all specific models: the broad DLTS line is caused by electron states filling a certain energy range, because one and the same defect is located in slightly different surroundings, which may be due to local strains, chemical inhomogeneity, or interaction. Variation of the filling pulse length tp brings to light another characteristic property of localized stages at EDs. The dependence on tp of the signal S (e.g., of the DLTS line), indicating that the occupation of the level is logarithmic: S - In tp. This can be understood as a consequence of capture over a barrier increasing with captured charge (Figielski, 1978). DLTS lines of dislocation-related centers always appear much broader than with isolated point defects. Formally this reflects a nonexponential transient of the capacitance during the emission period. In this context, an important paper by Figielski (1990) should be mentioned. This author considers Read's model (see the introduction to Sec. 6.3), but takes hopping of the excess electrons along the dislocation line seriously. First he calculates the extra energy that arises due to the nonuniform arrangement: A E = € , - E , = Nd f 2 E , a2 (where a is the standard deviation of the distribution of distances between electrons and Nd is the density of chargeable sites). Assuming a Gaussian distribution, he calculates the configuration entropy S, = Ndfk In [ a V @ 3 1 . Now the authors makes the important assumption that during electron emission


from the dislocation thermal equilibrium by rearrangement is preserved, Le., Gibbs free enthalpy G, is minimized. From this he obtains a relation between (J and f

(kT

)‘I2

(J=

(2Eo

f)

Now G, is known (for the case of most random arrangement)

kT G,=E,+-”,fln 2

(6-12)

For the virtual equilibrium transfer of an electron from the matrix to the dislocation, the change of G has to be zero (6-13)

+TAS+N-I~=O df

(where 5 is the distance between the conduction band edge and the Fermi level and AH - T AS is the change of free enthalpy by ionization of an occupied trap. The term M - E 0 f takes into account the diminishing of the ionization energy by rearrangement of the remaining electrons in the chain). This formula differs from Read’s minimum energy approximation by the last term. Solving the equation gives the occupation ratio in equilibrium fo. Now Figielski comes to his main topic, the kinetics of electron emission, i.e., the process that determines the DLTS signal. He defines the rate equation of the capture-emission process, i.e., an equation for dfldr. In thermal equilibrium, dfldr has to be zero. The main result of the calculation is

-d f dt

-(emission)

-f (6-14)

31 7

The author claims that the exponent 3/2, which can be traced back to the nonuniformity of the electron arrangement, “drastically modifies the kinetics of electron emission”. Figielski stresses that an activation energy with physical meaning in the present case can be obtained using the rate-window scanning technique. In contrast to point defects, an activation energy AH+Eo fo is obtained. A last remark about this paper concerns fluctuations of the ionization enthalpy, which must be connected with fluctuations of the distances between neighboring electrons. Figielski (1990) argues that the unusually large width of two (B and D) of the three main DLTS lines of deformed n-type silicon may well be due to those fluctuations, but not so in the case of line C (see later). Because of all these problems, the results of DLTS for deformed crystals at the present time can be taken only as preliminary. Nevertheless, by comparing widespread literature, some levels can be identified. DLTS of deformed n-type silicon to 10l6 cm-3 P) reveals three main peaks which each exhibit characteristic features. They are called B, C, and D following Omling et al. (1985). We characterize these lines by their position on the temperature scale (for a given rate window T-’) and by an apparent emission enthalpy AE, as found from the Arrhenius plot (Table 6-1). Common properties of these lines are: they are produced by plastic deformation, broadened, and exhibit logarithmic filling behavior. From this combination we can conclude that they are related to dislocations. It is interesting that B as well as D shows exponential filling (like isolated point defects) behavior for extremely short filling pulses ( t p Iio-’ s) (Omling et a].,

31 8

6 Dislocations

Table 6-1. The three main DLTS lines of plastically deformed n-type silicon. Line Position ( K ) at window ( 5 - l )

Apparent enthalpy (eV)

B

150

25

0.28

C,

200 226 260

38 38

0.44 0.505 0.71

Cz C

50

Ref.

Omling et al. (1985) Birkner (1994) Birkner (1994) Knobloch and Alexander (1996) (not conventional analysis)

1985). This confirms that logarithmic filling requires the presence of a barrier, which has first to be assembled. The most important problem concerns the nature of the dislocation-related defects: they may be located within the dislocation core of they may be point defects in the vicinity (and charge cylinder) of dislocations. One criterion can be thermal stability. It is well known that annealing at 800°C ( 1 5-30 min) greatly diminishes the number of point defects, not changing the dislocation density. In fact, this annealing removes lines B and D, whereas the two C lines (and the background C’ to be discussed below) stay essentially unchanged (Birkner, 1994). (This gives the opportunity to investigate C isolated.) In view of this result, the defect producing line C seems to be most closely related to dislocations*. It has been investigated in most detail using deformed n-type silicon which had been furnished with dislocation sources by scratching the surface and (partly) annealed after deformation (Kronewitz, 1991; Birkner, 1994). * The widely used RCA cleaning procedure (Kern et al., 1970) influences the DLTS lines of plastically deformed silicon in a different way; the C line is most resistent (Langenkamp, 1985).

Simulation of line C using the density of states corresponding to (AE)-3 reproduces the general shape of the line, but the lowtemperature side is too steep. To get better results, a second (AE)-3 distribution is necessary. Perfect fittings needs three Gaussian lines: the two lines C, and C2, which can be noticed directly (Omling et al., 1985), and in addition a rather wide background line C’ (full width half maximum FWHM = 200meV) (Birkner, 1994). At 800°C, all three components slowly anneal out, keeping their ratio constant. Schroter et al. (1995) came to the conclusion that C belongs to the class of localized states, i.e., that C is due to point defect clusters in the vicinity of dislocations (outer radius of the order 1.1 nm). Also, Cavalcoli et al. (1997) came to the conclusion that it is line C that is most closely related to dislocations. These authors only found the line consisting of two components (C,, C,) for a weakly (Le., within a short time) deformed specimen. In strongly deformed samples (dislocation density N 2 lo7 cm-2), one line C is found (apparent enthalpy 0.4 eV) which exhibits Gaussian, i.e., symmetrical, broadening. The amplitude of this line depends on the length of the filling pulse t, as well as on the emission rate e,. The first dependence is used to follow the formation of a Coulomb barrier (of the order of 0.2 eV when fully developed), the second is attributed to the influence of the peak temperature on the number of electrons overcoming this barrier. Despite the somewhat different experimental findings, the authors came to the same conclusion as Schroter et al. (1995), ascribing the DLTS line C to point defects in the vicinity of dislocations. Relating the number of C-traps to the dislocation density, they calculated the line density n of C-centers along the dislocations for high dislocation density ( N > 3 x IO6 cm-*) as n = 3 x


lo5 cm-’; for smaller N , the density n increases up to lo7 cm-’, not far from the numbers determined by Fell et al. (1993) from EBIC contrast (see later). For a long time, line D had been known to exhibit a peculiar property: decreasing the temperature of the peak decreases the line by factors exceeding two. This decrease appears most pronounced if TSCAP is used around 200 K (Kisielowski and Weber, 1991). Knobloch and Alexander (1996) showed that the D-center is amphoteric, exchanging electrons with the conduction band (e,) and holes with the valence band (ep); as soon as such a center lies not exactly mid gap, the DLTS signal becomes temperature-dependent, the temperature dependence being weighted with the ratio of the two capture cross sections

NT t A~ T exp - N~ en t = -ND

ND

’ 4 -N,

the logarithmic filling behavior are contradictory. The position of the center below mid gap (AE = 0.71 eV) suggests exchange with both bands; NT = 2 x lo’’ cm-3 is in agreement with the evidence for some 1015cm-3 point defects in the lower half of the gap of deformed silicon by photo-EPR (Erdmann and Alexander, 1979). The capture cross sections for electrons and holes, respectively, turn out (from A) three orders of magnitude different; but as usual their absolute values appear too large. From the filling characteristics at very short pulse length (Ilo-’ s), a, = 6 x cm2. The DLTS line B has not up to now been investigated in detail. It is broadened asymmetrically. Deformed p-type silicon has been studied less extensively than n-type Si, in spite of the fact that the lower half of the gap contains about five times more traps than the upper half. The spectrum of the hole traps consists of a broad group of overlapping peaks (Kimerling and Patel, 1979). The total density of related centers ( = 3 x lo” ~ m - is~ in) good agreement with that of EPR centers with long spin lattice relaxation time (Le., point defect clusters). After annealing, a broad peak at H 0.33 eV remains, which is ascribed to the dislocation by some authors. Photo-EPR showed that 1 ~ ~ can be exabout 2.5 x 1015~ 1 3 electrons cited from the energy range from 0.82 to 0.66 eV below the conduction band. Thus DLTS and EPR are in agreement as to the location of the number of hole traps (acceptors) in that energy range ( E = 2.7%). If we take thermal stability as indicative for correlation of the particular defect with dislocations, the electron trap C and the hole traps around E,+0.33 eV are most suspicious. In this connection, it is worth having a look at the doping dependence of the intensity of the EPR spectra (Kisielowski et al.,

(3 (6-15)

with

up

Nc On

where AE is the distance of the level from the conduction band, Eg the gap width, NT the density of the centers, a, the capture cross sections, N , the density of states at the valence band edge, z the inverse rate window, andA,= a,N,). Perfect simulation of the temperature dependence of the line is possible over a wide range of rate windows using the rate equation for electron and hole emission without a barrier term; however, Gaussian broadening of the energy level (full width half maximum of 54 meV) is implemented. In the framework of the present interpretation of DLTS, the lack of a barrier term and

31 9

320

6 Dislocations

1991). If we assume that plastic deformation generates the same types of defects in the doping range from 10’’ cm-3 p type to l O ” ~ m - n~ type, we can determine the position of the Fermi level at which the defects enter or leave the paramagnetic charge state. Again there are some problems limiting the accuracy of the method because of the high conductivity of the doped crystals at room temperature. The point defects, normally detected at room temperature in slow passage, have to be recorded in adiabatic fast passage at 24 K. The dislocationrelated centers Si-Y and Si-K 1/K 2 are measured in adiabatic rapid passage at 8 K. The results are as follows: the spin density of both groups of paramagnetic centers decreases markedly beyond a critical doping of 3 x 10ls cm-3, by phosphorous and by boron. Within those limits the behavior of point-like and dislocation-related defects is somewhat different: The density of point defects in the paramagnetic charge state decreases from p doping over undoped material to n doping: in contrast, the density of the dislocation center Si-Y stays constant. The interpretation is as follows: All deformation-induced centers may assume at least three different charge states (Fig. 6-12); only when the Fermi level lies between their 0/+ and 4 0 levels they are paramagnetic. Apparently,

0

r,

....+,

I.....

v

Figure 6-12. Energy levels of an amphoteric defect (schematic). If the Fermi level is in the middle range, the defect is neutral. The defect will be EPR active only in one charge state. C, V: edge of the conduction and valence band, respectively.

the Fermi level for most defects near a doping of +/- 3 x 1015cmW3passes through those levels. For the point defects the positions of those levels will be somewhat different depending on type; the decrease of the total number when E, passes from the lower half of the gap into the upper half reflects the fact that more types of point defects centers in fact have their characteristic levels in the lower half. Unfortunately, in deformed crystals the (local) Fermi level cannot be calculated from the doping but depends in a complicated manner on all the defects, including their spatial arrangement. We will now comment on the investigation of dislocations in semiconductors by the EBIC (elelctron beam induced current) technique. We start with the general observation that dislocations act as acceptors for majority carriers, Le., they are negatively charged in n-type semiconductors and positively charged in p-type material. The surrounding screening charge has the respective opposite sign. As explained earlier the energy level of a dislocation in n-type material increases with line charge qL. This increase may end in two ways: either all the chargeable sites (density Nd) are filled before the level reaches the Fermi level (EF), or the level coincides with EF when only part of Nd is filled. In the first case (l), enhancement of the temperature T (i.e., lowering of EF) in the first instance has no influence on the charge of the dislocation. However, when E, reaches the dislocation level it takes this level with it (level pinning), so decreasing the charge and the barrier. In the second case (2), pinning prevents the dislocation from ever capturing the maximum possible charge (q Lmax = Nd e ) . EBIC consists of the injection of electron-hole pairs by a high voltage electron beam. The charged dislocation accumulates minority carriers which reduce the line

6.3 Experimental Results on the Electronic Properties

charge qL (and the barrier). A new equilibrium is reached when as many majorities by thermal activation reach the dislocation as minorities diffuse to the dislocation. This new equilibrium charge is smaller than the charge before irradiation. The reduction is larger the lower the temperature, because the barrier must be lower at low T. In other words, the equilibrium charge under the beam increases with increasing temperature. Eventually, any further increase is stopped and the charge begins to decrease on further raising the temperature. The charge as a function of T passes through a maximum, which takes the shape of a plateau in the first case (1). Wilshaw and Fell (1989) showed that the EBIC contrast (c) in good approximation is proportional to the line charge q L . Therefore, the contrast is expected at first to increase with increasing temperature. In a similar manner, it can be deduced that there is a decrease of the contrast with increasing excitation (beam current). From measurements of EBIC contrast as a function of temperature and beam current, either Nd(1) or the position E, of the (neutral) dislocation (2) may be deduced (Wilshaw and Fell, 1989). It may safely be said that the predictions of the theory of Wilshaw and co-workers (1985, 1989) have been found to agree with the result of many experimental investigations, even though a new class of EBIC contrasts of dislocations has recently been found (see Sec. 6.3.3.5). In a recent paper (Fell et a]., 1993), some quantitative results on dislocations in plastically deformed n-type silicon were reported based on Wilshaw’s theory. Fell and co-workers compared “clean” dislocations with copper-contaminated ones. After deformation at 650 and 900°C under clean conditions, the recombination-active level is pinned to EF, i.e., case 2 is realized. The

of Dislocations

32 1

level (we call it a) position turns out to be beween 0.3 1 and 0.39 eV below the conduction band edge E,. (The value increases with temperature, indicating a bundle of states of a certain width.) The lower limit of the density Nd of chargeable states along the dislocation is 2 x 10’ m-’. After deformation at 420”C, case 1 is found: a much deeper level ( y ) (E, > 0.52 eV) stays below E,. Now a definite value Nd can be determined ( 2 . 8 - 2 . 9 ~ lo8 m-’). If we now look at the results with copper-decorated dislocations, we find for Tdef= 650°C a level fl (deeper than a) (E, = 0.5 eV), while for Tdef= 420°C the contamination does not make any effect. The interpretation of these results is as follows: clean dislocations of different structure (dependent on deformation conditions) are characterized by the levels a and y, respectively, while copper (in or near the dislocation) produces p. For the 420 “ C dislocations, the deeper level is in both cases y (with and without copper), so that p even if it is present (with copper) is lifted above E,, being electrically inactive. Typical values of contrast in this work are c = 1.5% (if level a is concerned) and c = 5 -6% (levels p and y, respectively). Because EBIC contrast c indicates shortening of the minority lifetime, this quantity is a measure for the electrical activity (as a recombination center) of a given defect. Bondarenko et al. (1996) tried to find out the distribution of recombination centers around dislocations in silicon by analyzing the EBIC contrast profile. From the dependence of the width of the EBIC profile on the injection level, the authors concluded that the prevailing recombination mechanism is via the atmosphere of point defects around the dislocation. The authors solve a system of equations for diffusion, drift and charge collection probability to get the distribution of recombination events by fitting

322

6 Dislocations

to the measured profile. They obtain a characteristic range of active centers R, = 1.31.8 pm *. The density of the centers is esticm-3 and the capture mated as N , = cross section as oP- 5 x 1 0-13 cm2. This review on EBIC investigations is based on Wilshaw's assumption that the recombination of minority carriers should be controlled by the charge of deep states and is called in the literature CCR (charge controlled recombination) theory. There is a second class of EBIC contrasts with different contrast behavior with respect to temperature and beam current (Kittler and Seifert, 1993a). Because these cases are explained by shallow dislocation related states, they are treated i n Sec. 6.3.3.5. Here it should be pointed out that some authors doubt the existence of charge and energy barrier under the electron beam. Kittler and Seifert (1994) show that both classes of contrast behavior can be deduced from the Shockley-Read-Hall (SRH) theory of recombination at isolated point defects (see Sze, 1985) if for class 1, which can also be explained by CCR theory, deep recombination centers are assumed with shallow centers for class 2 . Until the application of SRH theory is quantitatively carried out for a greater number of cases, the question is open for discussion. For a more detailed discussion, see Holt (1996) and Kittler and Seifert (1996). In any case, i t is a priority to examine the persistence of the dislocation charge under an electron beam by computer simulations (see Kaufmann and Balk, 1995).

* In spite of the qualitative agreement between this result and the conclusion drawn by Schroter et al. (1995) from DLTS spectra, it has to be noticed that the point defect cloud here is three orders of magnitude larger than in Schroter et al.'s case. These authors claim that clouds that are generated by the strain field of dislocations have at most a few nanometers radius.

6.3.3 Phenomena Indicating Shallow Dislocation-Related States 6.3.3.1 Photoluminescence (PL)

In 1976 Drozdov et al. were the first to show that silicon crystals containing dislocations exhibit 4 PL lines (D1 to D4) with photon energies between 0.812 eV and 1 .OOO eV. Sauer et al. (1 985), investigating the response of the spectrum to uniaxial stress, concluded that D 1 and D 2 in fact are to be ascribed to point defect centers with their (tetragonal) (100) axis in random orientation. D 3 and D 4 on the other hand appeared to be correlated to dislocations. This is most convincingly shown by the modification that the line pair shows when the dislocation morphology is changed by high-stress-low-temperature deformation (Sauer et al., 1986). As mentioned earlier, this deformation procedure results in straight dislocations, parallel to three (1 10) directions; the percentage of screws is greatly increased and the dissociation width d of all dislocations is changed, i.e., partly increased and partly decreased (3 nm Id I 12 nm). Instead of the lines D 3 and D4, crystals with this dislocation morphology show a new spectrum (D5) consisting of a series of narrow lines with phonon replicas (Fig. 6-13). Very weak annealing (200 to 360°C) transforms this spectrum back into D4 (part of D 3 turning out to be a phonon replica of D4). By this annealing, d is relaxed to its equilibrium value d, ( 5 nm). Consideration of the reaction kinetics of the new spectrum led Sauer et al. (1986) to assume donor-acceptor recombination as the actual type of PL. It could be shown that each of the lines of the D 5 series corresponds to a certain value of d in correspondence to the periodicity of the Peierls potential. Sauer et al. (1986) proposed transition between donors at one partial dislocation and acceptors at the other as the physical na-


Photon energy (eV1 0.90 0.95 '

)

,

I

1

1

1

1

1

323

1.00 1

1

1

1

1

Figure 6-13. Silicon photoluminescence. The crystal is two-step deformed. (a) Excitation dependence; (b) change of the spectrum during isochronal annealing ( 1 h). TO, TA: phonon replica (from Sauer et al, 1986).

ture of the recombination process, the interaction between the two depending on the distance d between the two partials. Later on, a similar spectrum was found in high-stress-deformed germanium (Lelikov et a]., 1989). Using special compression axes recently, the dissociation width d of the dislocations could either be increased or decreased without moving the dislocation as a whole. In this way, it could be demonstrated that the lines fan out to the high-energy side when d decreases and vice versa (Izo-

tov et al., 1990). This is confirmation for another model for the particular PL as recombination of one-dimensional excitons of the Mott type bound to the core of 90" partial dislocations. The energy of those excitons will be influenced by the strain field of the 30" partial completing a 60" dislocation. Lelikov et a]. (1989) estimate the binding energy of electron and hole in deformation potential bands accompanying the dislocation line as 150 and 80meV, respectively (germanium). Be that as it may, comparing

324

6 Dislocations

the photon energies with the width of the band gap clearly prooves the existence of shallow states near or in the dislocations in silicon and germanium.

6.3.3.2 Optical Absorption After plastic deformation, the optical absorption in front of the fundamental absorption edge is clearly increased for silicon and for gallium arsenide. Bazhenov and Krasilnikova ( 1986) calculate local band gap narrowing in the strain field of dislocations (deformation potential). The temperature dependence agrees well with the absorption spectrum observed with GaAs (gap narrowing by 200 meV), but for silicon this is not true before thermal annealing of the crystals at 800°C. Then the gap is narrowed by 170 meV.

6.3.3.3 Microwave Conductivity (MWC) One of the most attractive ideas i n dislocation physics is to analyze (and possibly to use) dislocations as one-dimensional wires of high (metallic?) conductivity embedded into a matrix whose resistivity can be chosen and controlled. By the way, D. M. Lee et al. (1988) are realizing this idea by decorating straight and parallel misfit dislocations with nickel, but here the dislocations are only nucleation centers for a second phase. We here focus on the ability of clean dislocations to carry current (dc or ac). A multitude of papers has appeared over the years in this field. The most recent work taking into account both the important role that point defects play and the possibility to completely change dislocation morphology is that by Brohl and co-workers concerning FZ silicon (Brohl and Alexander, 1989; Brohl, 1990; Brohl et al., 1990).The authors

used pre-deformation at 800°C ( E = 1.6%) followed by an annealing step (16 h) at the same temperature. As shown by EPR and DLTS, no point defect (PD) centers can be found in those crystals. Subsequently the dislocations are straightened parallel to (1 10) by a short (30’) deformation at 420°C. This deformation step produces only a few PD’s because the dislocations are moved only over a short distance. An influence of these remaining PD’s can be demonstrated by MWC under monochromatic illumination at a doping of 5 x 1014~ m - but ~ , it is very weak at 4 x 1015cm-3 (“effective doping”). The pronounced anisotropy of the dislocation morphology makes it easy now to demonstrate that MWC is parallel to the dislocations. This MWC (frequency 9 GHz) can be separated from the bulk contribution below 20 K, where it dominates by several orders of magnitude (Fig. 6-14). Relaxing the high-stress morphology of the dislocations reduces the MWC and leads back to the predeformed state of both the anisotropy and the size of the effect. Of key importance for understanding the MWC is the observation that a certain doping (n- or p-type) is necessary. Strictly speaking, MWC depends on effective doping, i.e., excess of chemical doping over the amphoteric effect of deep point defect levels. That (only) electrons or holes captured by the dislocation from dopants cause MWC had been shown before by Ossipyan ( 1 985) i n germanium by an elegant experiment: Neutron transformation doping produced (randomly distributed) dopant atoms - first gallium acceptors and then arsenic donors. Conductivity is developed exactly parallel with the actual effective doping. Discussing his results, Brohl (1990) ascribes MWC to band conductivity in shallow bands near to the conduction and valence band edges. The author first excludes hopping conductivity

6.3 Experimental Results

on the Electronic Properties of Dislocations

0 0 0 0 0 0

>

0

0

0 0

0 0

0 0

0

Figure 6-14. n-type (4.4 x 1015 cm-3 P) silicon, microwave conductivity. (a) Lower curves: only predeformed at 800°C and annealed at 800°C, 16 h. Upper curves: additionally deformed at 420°C under high stress. 0:electric field parallel to screw dislocations. A, x: electric field parallel to edge dislocations. (b) Two step deformed crystal rotated around the Burgers vector of the dislocations. Maximum conductiviy when the field is parallel to the glide plane (from Brohl, 1990).

along an impurity band by a quantitative argument: even if all doping atoms were collected by dislocations, their mutual distance, would be much too big for hopping. Next, it is suggested that the dislocation may be connected with two bands; one is near the valence band and is full when the dislocation is neutral, and the other is near the conduction band and is empty. Conductivity is produced when the lower band accepts

325

holes, or the upper band electrons. Considering the latter case, one may calculate the maximum line charge by assuming that all electrons which are lacking in the EPR of (neutral) phosphorous are accepted by dislocations. For a certain experiment qL was qL5 1.4 x lo-' As m-l (corresponding to 8.5 x lo5 electrons per cm). The respective band bending is 120 meV. Because conductivity was observed, the Fermi level (coinciding with the phosphorous level E,45 meV) must be inside the dislocation band. This means that band bending by at most 120 meV lifts the dislocation band onto the phosphorous level, so that the distance from the conduction band is at most 165 meV. A corresponding result is reached for the donor band near the valence band. Influence of illumination by monochromatic light establishes the position of the deep levels competing with the dislocation band for the electrons from the phosphorous atoms: occupation limits of those states are 0.6 and 0.85 eV below the conduction band. MWC by screw and 60" dislocations is of comparable magnitude. This, in our view, supports the idea that MWC is a matter of the elastic strain field and not of the dislocation core. Because dislocation segments of limited length are probably the conducting elements, it is extremely difficult to extract quantitative data on the conductivity of a single dislocation. A lower limit for the electron mobility of 100 cm2 Vs-' was estimated. The authors believe in the deformation potential of the strain field of the dislocations. But also lateral confinement of the carriers by the potential wells accompanying charged dislocations can induce local hole bands at the top of the valence band and onedimensional resonant states below the conduction band (Fig. 6-9).

326

6 Dislocations

6.3.3.4 Electric Dipole Spin Resonance (EDSR) Kveder et al. (1984, 1989), investigating deformed n-type silicon, found transitions between Zeeman terms induced by the electrical component of a microwave field (9 GHz). This type of combined resonance is due to spin-orbit interaction and also depends on high (band) mobility of electrons. A further requirement is a strong crystal field. In a cubic crystal the field is restricted to the core region of symmetry-breaking dislocations. That band mobility is under consideration can be seen from the fact that the effect is observed at 1.4 K. The EDSR is characterized by a special anisotropy from which the direction of the carrier motion can be deduced. In n-type silicon only one direction was found corresponding to the orientation of Lomer dislocations. The electrons were ascribed to a band 0.35 eV below the conduction band. Recently, EDSR was demonstrated also in high-stress-deformed p-type silicon (Wattenbach et al., 1990). In contrast to ntype crystals, up to 9 lines are observed in this case. The analysis is not yet complete, but apparently dislocations of several orientations are active. Thus we suppose that the shallow bands established by MWC and connected with 60" and screw dislocations are responsible for the effect. This would mean that relatively widely extended wave functions (perpendicular to the dislocation line) feel enough spin orbit interaction to produce a very strong effect. Coming back to n-type silicon, a detailed investigation has been carried out, making sure that the overwhelming majority of dislocations were straight (parallel to the three (1 10)directions in only one glide plane) and with the dissociation width either enlarged or reduced (high stress deformation) (Wattenbach and Alexander, 1993). The spec-

trum (called Si-K 10) is proven to be EDSR by the line shape, the dependence on the electric microwave field, and the extreme anisotropy of its amplitude; moreover, it deand pends on effective doping IN,-N,I light, in the same way as microwave conductivity along dislocations (Sec. 6.3.3.3), where the role of deep acceptors is taken by the point defect clusters produced by plastic deformation. It is interesting to compare that anisotropy with the spectrum Chi found by Kveder (Kveder et al., 1984, 1989), which we could reproduce by low-stress deformation. Chi can be explained by the quasi-classical theory outlined earlier, assuming spin-flip of electrons oscillating along Lomer dislocations. In contrast, the only strong K 10 line has two zero orientations 60" apart. Formally, the strength of the line for several rotation axes can be described by electron motion along the (widely split) 90/30 dislocations, being zero when the electric microwave field is parallel to one of the two partial Burgers vectors (the latter property not yet being understood). Different behavior of electrons moving in shallow bands along Lomer dislocations and dissociated 60" dislocations, respectively, can be expected, since Lomer dislocations are not split into partials (Bourret et al., 1983). As mentioned earlier (Sec. 6.3.1 j plastically deformed germanium with a reduced abundance of the isotope 73Ge also exhibits an EDSR spectrum. (Pilar v. Pilchau et al., 1992).Due to the predominance of one glide plane (1 i l ) , only the related line group out of the four found by Pakulis and Jeffries (1981) is present. Again, the roles of the electric microwave field strength and of sub-bandgap light indicate EDSR. The position of the line on the magnetic field scale yields the g-tensor of the resonance center. It is rather near to the theoretical prediction (Roth and Lax, 1959, 1960) for electrons


moving on one of the four ellipsoidal energy surfaces (“valleys”) of the conduction band of germanium. Our thesis is: we observe (electrically induced) spin-flip of electrons moving in a shallow band split from one of the conduction band minima by the strain field of the dislocations in the (primary) glide plane (Winter, 1978).

6.3.3.5 Electron Beam Induced Current (EBIC) In 1992, Kittler and Seifert (1993 a) demonstrated that the EBIC contrast of dislocations can depend on temperature (and beam current, respectively) in two distinctly different ways. The authors proposed the following classification: Dislocations of group 1 behave as explained by the theory of Wilshaw and Fell (1989): the EBIC contrast increases with increasing temperature and decreases with increasing beam current. Members of group 2 show essentially the opposite behavior. Strikingly two neighboring dislocations in the same specimen can belong to different groups (Kittler and Seifert, 1993b). Kittler and Seifert (1993 a) present a collection from literature which shows that dislocations in silicon, both of n- and p-type, as well as in n-GaAs, can behave in both ways. The authors deduce group 2 properties from Shockley-Read-Hall recombination theory assuming shallow recombination centers for minority carriers. Bondarenko and Yakimov (1990) demonstrated by measurements using a metal microprobe on dislocations with group 2 behavior in annealed Czochralski-grown silicon, the absence of an electrostatic barrier, Le., of line charge. Considering the calculations by Kittler and Seifert (1 993 a), it is concluded that pronounced group 2 behavior suggests states in or near dislocations that are 70100 meV from the next band - in agreement

327

with the results obtained by other methods described in this section.

The Role of Contamination of Dislocations by Metal Atoms It has been discussed repeatedly in the past whether the physical properties of dislocations in semiconductors might be influenced by unintentional decoration with impurity atoms. Those impurities are expected to be trapped in the strain field of the dislocation or precipitated as a new phase along the dislocation line. Recently first clear experimental results were published, mainly with respect to the influence of transition metal atoms on dislocation-related photoluminescence (PL) (Higgs et al., 1990a) and EBIC contrast (Higgs et al., 1991; Wilshaw, 1990) in silicon. The authors investigated dislocations in epitaxial layers and in crystals deformed under extremely pure conditions without detectable contamination (c 10’’ ~ m - before ~) and after contamination by back plating with Cu, Ni and Fe, respectively. In the “pure” state no PL nor EBIC contrast of dislocations could be detected. Materials with low levels of copper ( ~ 1 0 ~’ m ~ - showed ~) both PL and EBIC contrast strongly. Interestingly, the three metals investigated exhibited almost identical effects. Considering TEM analysis there are two regimes of contamination: below and above one monolayer of metal atoms on the surface, respectively. Only in the second regime can precipitates be seen at partial dislocations. Both PL and EBIC contrasts increase with contamination up to 0.1 monolayer. Further enhancement of decoration destroys PL (radiative recombination), while EBIC contrast persists and often increases up to dislocations with precipitates. The authors emphasize that various models for direct or indirect (via other dislocation-related point defects) effects of metallic impurities are conceivable.

328

6 Dislocations

Peaker et al. (1989) analyzed the effect of gold and platinum on oxygen-induced stacking faults (SF) surrounded by Frank partials. The basic material was vapor phase epitaxy silicon layers with an extremely low concentration of electrically active defects. After generating the SFs under clean conditions, the PL line D1 (cf. Sec. 5.3.1) and, by DLTS, a deep electron trap (activation enthalpy 0.415 eV) have been observed. After contamination with gold or platinum, D1 exhibited some modification of its shape and the trap position shifted gradually to midgap, irrespective of the fact that Au and Pt, respectively, produce as point defects quite different traps. At the same time the capture characteristics of the SF-related trap changed from being point-defect-like to logarithmic, which is typical for extended defects. From the gradual change of the trap position the authors conclude that the traps present at the clean Frank partials (415 meV) are modified by additional electrically active centers. The density of those centers is rather low (a few per 100 A dislocation length). These and other results raise the fundamental question whether clean dislocations in silicon are electrically active at all. Even such processes as dislocation motion and multiplication seem to be influenced by decoration, since Higgs et al. (1 990 b) found the generation of misfit dislocations in strained layers of Ge,Si,-, to be strongly influenced by the presence of copper. Because extreme care is required to avoid any noticeable contamination of silicon by transition metals (Higgs et al., 1990a) one can assume that most experiments and applications concern at least lightly decorated dislocations. Thus, i n order to obtain a better understanding of clean dislocations, further clarification of the role of low concentrations of impurities is of great importance. And, it can already be concluded now that

some lack of reproducibility typical for measurements on dislocation properties may be traced back to these effects.

6.3.4 Germanium The Hall data indicating the presence of a half-filled band 90 meV above the valence band in p-type germanium were mentioned at the beginning of this section. A DLTS study (Baumann and Schroter, 1983 a) of deformed p-type germanium reveals four wellresolved peaks to be attributed to deformation induced defects. One of them, exhibiting logarithmic filling behavior and an activation enthalpy of the emission of holes of 0.075 eV, is tentatively ascribed to the dislocation band at (E,+0.09 eV) deduced from Hall data. Two other DLTS lines (H 0.27 eV, H 0.19 eV) are saturated by the shortest filling pulse and are therefore interpreted as isolated point defect centers. The fourth DLTS line (H 0.39 eV) is only found when the deformation temperature was below 0.6 T,. Both this thermal instability and the filling characteristics proove that the center also is an isolated point defect. Its capture cross section is a, = 1 0 - l ~cm2, and its density was determined as 1 . 4 ~ 1013cm-3 for a dislocation density 3 . 3 ~ lo7 cm-*, which corresponds to a calculated strain = 6 ~ 1 0 - ~ . In n-type germanium (Baumann and Schroter, 1983 b), a single broad and asymmetric DLTS peak exists after deformation at 420°C. Annealing at 580°C reduces the amplitude by an order of magnitude. The trap position is (E,-0.20 eV) - coinciding with the position of the hole trap H 0.39 eV. After annealing, the filling behavior clearly shows this electron trap to be an isolated point defect; before annealing, the filling behavior is complicated, probably indicating that most of the defects here are under the influence of the barrier of charged dis-


locations. It is this part of the defects which anneals out. Also, Hall effect data (Schroter et al., 1980) indicate that the density of holes suddenly increases markedly when the deformation temperature is lower than 0.6 T , (of course, the critical temperature depends somewhat on the cooling procedure applied after deformation). On the whole, the findings on the defect spectrum after deformation for germanium are similar to that of silicon: several deep traps are present, with more in the lower half of the gap. Most of them - if not all - are not directly dislocation related. Above 60% of the (absolute) melting temperature, a large portion of the point defect centers becomes unstable. Dislocations give rise to a shallow band.

6.3.5 Gallium Arsenide One of the main problems for applying semi-insulating GaAs for high integration of semiconductor devices is spatial inhomogeneity of electrical parameters over the wafer, often correlated with the distribution of dislocations. Whether this is due to interaction between dislocations and point defects or to electrical properties of the dislocations themselves, it has induced strong research activity on dislocations in plastically deformed gallium arsenide. Proceeding from elemental semiconductors to compounds, there is an increase in the number of material parameters which have to be known if results from different authors are to be compared: the crystals are grown by different techniques, producing quite different combinations of defects: there are indeed more types of defects, the most prominent among the new ones being the antjsite (AD) defects AsGa and GaAs. Also, the number of dislocation types has doubled (Comp. Sec. 6.2).

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A critical review of the literature gives the impression that all authors agree that the most striking effect of plastic deformation is a strong decrease in the density of free electrons in n-type GaAs (Gerthsen, 1986; Suezawa and Sumino, 1986; Skowronski et al., 1987; Wosinski and Figielski, 1989), which is proportional to the plastic strain (An = -(2-6xI0l6 ~ m - for ~ ) E = 3%). Optical DLTS locates these acceptors at about E,+0.45 eV (Skowronski et al., 1987). Other authors using other methods arrive at E,+0.38 eV (Gerthsen, 1986) and E, + 0.37 eV (Wosinski, 1990), respectively. The acceptors compensate shallow donors in n-type material, but they are too deep to significantly influence the density of free holes in deformed p-type GaAs. In semi-insulating (si) GaAs they lower somewhat the Fermi level, transforming part of the EL2 double donor into the paramagnetic charge state (EL2)+. In spite of enormous efforts, the atomistic structure of EL2 is not yet established, but according to common belief, it contains an anion AD AsGawhich in turn produces a characteristic four-line EPR spectrum. The increase of this spectrum with strain E was previously explained as proof for the generation of EL2 defects by moving dislocations. Later on, some doubt arose, because the additional defects did not exhibit the quenchability by light which is characteristic forEL2 (Omlinget a]., 1986). Thus the present view is that the shift of the Fermi level caused by the deformation-induced acceptors changes only the charge state of part of the EL2 defects present before the deformation. However, this does not seem to be the complete answer either: Fanelsa (1 989) recently deformed highly n-doped (1.8 x 1017cm-3 Si) GaAs which, as prooved by DLTS and known by others (Lagowski and Gatos, 1982), is free of EL2. After plastic deformation by 3% at 400°C, DLTS showed 1 . 3 ~ 1 0cm-3 ’ ~ EL2 defects.

330

6 Dislocations

In view of the inhomogeneous compensation, this is a lower limit. So we think that both effects - ionization and generation can be responsible for the increase of the AD spectrum, and each case is to be analyzed separately. No (other) donors are found after 3% deformation by Skowronski et al. ( I 987). Wosinski (Wosinski and Figielski, 1989; Wosinski, 1990), also using DLTS, was able to isolate two new traps occurring only after plastic deformation: an electron trap ED 1 (E,-0.68 eV, D = cm2) and a hole cm‘) trap HD 1 ( E , + 0.37 eV, D = (Fig. 6-15). Both traps have in common a logarithmic filling characteristic with variation of the filling pulse length over a range of six orders of magnitude. This is the reason why only a lower limit for the trap density can be given: 2.5 x l O I 5 and 2 x l O I 5 cm-3 for a - calculated - dislocation density of 8.5 x 10’ cm-2. The authors conclude from the rigorously logarithmic filling and from the proportionality of the trap and dislocation densities that these traps belong to dislocation cores ( a and p dislocations?). It has been shown that ED 1 acts as recombination center for photo-excited carriers. Two optical effects are assumed to be strongly related to dislocations. The first effect involves the absorption tail near the fundamental absorption edge, after subtraction of the contribution by intracenter transitions in EL2 centers. This absorption tail is found by several authors (Bazhenov and Krasilnikova, 1986; Farvacque et al., 1989; Skowronski et al., 1987) and is explained by gap narrowing due to elastic strain or electric fields (Franz-Keldysh effect) near charged dislocations. From comparison of this tail i n n- and p-type GaAs, Farvacque et al. (1989) deduced the existence of two dislocation “bands”, one near the valence band, the other at midgap. Extending the considera-

/

0

100

I

150

200

250

300

Temperature IK)

I

(b)

Temperature [K)

Figure 6-15. Gallium arsenide, DLTS spectra. (a) ptype GaAs dashed: undeformed; solid: plastic strain 2 8 (rate window: 17 s-’, filling pulse: 1 ms (Wosinski, 1990)). (b) n-type GaAs (LEC). dashed: undeformed; solid: plastic strain 2.8%. Rate window: 5 s-’, filling pulse: 5 ms (from Wosinski, 1990).


tion to the photoplastic effect (REDG: radiation-enhanced dislocation glide), the authors hesitate to think of true (delocalized) band states, since the recombination process contributing part of the activation energy of dislocation motion should occur at a certain local center. But the existence of a space charge with electrical field is not restricted to a continuous line charge. The second optical effect is a photoluminesence band at 1.13 eV photon energy detected by Suezawa and Sumino (1986) and Farvacque et al. (1 989). But the authors make contradictory remarks on the necessary doping. Finally, we should mention that plastic deformation of GaAs brings into existence a second four-line EPR spectrum starting at E = 4% (as the AD AsGa spectrum) and then increasing linearly with strain to about 1017cmP3at 10% strain (Wattenbach et al., 1989). The anisotropy of the center is much more complicated than that of AsGa. Christoffel et al. (1990) attribute the spectrum to two trigonal arsenic interstitial complexes. We found the same spectrum in plastically deformed Gap, also (Palm et al., 1991). Because Ga is the only species (in high enough concentration) with nuclear spin 3/2 we feel sure that we are observing the spectrum of the AD GaAs. The low symmetry may be due to Jahn-Teller distortion (Kriiger and Alexander, 1991). To summarize: in plastically deformed GaAs comparable concentrations of both types of antisite defects are present. 6.3.6 A"BV' Compounds The compounds of Zn and Cd with anions from group VI of the periodic table (0,S, Se, Te) * are unique among the semiconductors with respect to electrical effects of plas-

* There are only seven such compounds, since CdO is an NaCl type compound.

331

tic deformation. The crystal structures (cubic zinc blende = sphalerite and hexagonal wurtzite), both are generated from tetrahedral groups of atoms with pure heterocoordination. The dislocation geometry is the same as in A"'BV compounds with the only exception that in wurtzite-type crystals, only one close-packed glide plane (0001) exists instead of four in zinc blende (1 1 1). But dislocations with an edge component in A"BV' compound crystals carry large electrical charges (up to one elementary charge per lattice plane). This leads to a number of closely related effects. The following survey is mainly based on the recent review by Ossipyan et al. (1986). Before some of these effects are described, the origin of the dislocation charge will be discussed. The first idea concerns the (partly) ionic bonding of the compounds: in the core of any (perfect or partial) dislocation, a row of ions of the same type will carry a net charge although not necessarily of the same magnitude as a row of the same ions in the undisturbed bulk because of possible reconstruction. (For the latter, charged crystal surfaces give some hints.) Detailed consideration shows that perfect 60" glide set dislocations carry charges of +(3 e*/4 b ) per lattice plane when e* is the effective charge of an ion in the lattice. With 0.28 I e*le 50.53 (Phillips and van Vechten, 1969), this results in inherent dislocation charges of (0.2-0.4) e/b for the series of compounds. The experimentally determined line charges for moving dislocations in the dark vary from 0.12 up to 0.7 e/b and are negative in n-type crystals throughout and positive in the only p-type material investigated so far (ZnTe). Thus the inherent ionic charge of the atoms in the dislocation core must be taken into account, but it cannot be a determinant for the dislocation effect. This is clearly demonstrated by the de-

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6 Dislocations

pendence of the line charge qL on illumination. The next assumption could be that charged point defects are collected by the dislocation when moving. This is excluded from consideration. the main reason also being the role of illumination. Free carriers swept along by the electric field of the moving dislocation can be neglected, since no screening of the line charge was observed. (It should be noted that Hutson (1983) takes a different view here.) Finally, carriers remain as a source of the dislocation line charge which are captured into deep trap levels connected with the dislocation core. In this case the sign of the charge is not necessarily related to the ionic type of the particular dislocation: rather, as shown by experimental findings, the dislocation seems to act as a trap for majority carriers, as is usual. Ossipyan et al. (1986), using accepted parameters, calculated the charge to be expected on the basis of the Read model for a perfect Se (g) 60” dislocation in ZnSe with 1015cm-3 shallow donors (-0.6 x lo-’’ C/m) and found it to be smaller than in experiments (-2xlO-’’ C/m). From this observation as well as others, the authors concluded that moving dislocations in A”BV’ compounds are not i n thermal equilibrium with the electronic subsystem as far as the occupation of the electronic states of the dislocations is concerned. They outline a model of a charged dislocation (the charge at the beginning of motion being the ionic charge) which on its way interacts with charged point defects. The dislocation may capture electrons (or holes) from those point defects, and on the other hand, electrons may (thermally assisted) tunnel through the barrier around the dislocation into the conduction band. During its path, the dislocation will assume a dynamical equilibrium charge (Kirichenko et al., 1978; Petrenko and Whitworth, 1980). It amounts

to qL= 1.67 x lo-’’ C/m for the case mentioned above. Comparing capture rate and emission rate, a logarithmic increase of q with dislocation velocity v is obtained; in fact, such a relation between qL and strain rate is found for strain rates exceeding 10-6 s-l. For smaller strain rates the theory is not applicable because here the screening charge (made from ionizd shallow centers) is believed to move with the dislocation. Experiments

The experiments which bring to light the dislocation charge (some of them indeed spectacular) can be divided into two groups: Either an outer influence (illumination, electric field, carrier injection) changes the flow stress of a deforming crystal, or during deformation electrical effects are induced as charge flow from one side face of the crystal to the opposite side face (“dislocation current”), change of the electrical conductivity or luminescence. We restrict ourselves here to a description of one experiment from each group. Charge flow connected with dislocations (Ossipyan and Petrenko, 1975) can be observed by macroscopic methods only if there are differences between the net charge transported by positive and negative dislocations. Since the sign of the charge is the same for the two types in A”BV’ compounds, only the difference of the size of the charge, or of the product from dislocation number times distance travelled by the average dislocation remains. Fortunately, the mobility of A (8) and B (8) dislocations mostly is extremely different, so that one type practically determines the charge flow. In this manner, it was possible to measure dislocation charges. Care must be taken in the case of low-resistivity samples where screening by mobile carriers can reduce the effective charge transported. On the basis of dislocation current, it can be easily under-


stood that immersing the deforming crystals into mercury will decrease the flow stress (by avoiding surface charges), and that application of an electric field to the side faces also influences the crystals' flow stress (electroplastic effect). The second key experiment to be discussed is the sudden and reversible change of the flow stress induced by illumination during deformation (photoplastic effect PPE (Ossipyan and Savchenko, 1968)). In most cases, the flow stress increases (up to a factor of two) - positive PPE - but there are also cases of negative PPE. Takeuchi et al. (1983) gave an extended review on the PPE in A"BV' compounds. The nature of the PPE is by no means clear, and we will come back to it in Sec. 6.5.5. However, there is no reasonable doubt that illumination changes the dislocation charge. It was shown that the spectral dependence of both the dislocation charge qL and the PPE are virtually identical; maximum effects are produced by photons just below the respective band gap energy. Moreover, there is a linear relation between qL and the flow stress (Petrenko and Whitworth, 1980). Deformation -Induced Luminescence

CdS and CdSe, in contrast to other semiconductors, are ductile down to liquid helium temperature. This makes those compounds very attractive for optical in-situ analysis of the deformation process. Weak thermal fluctuations prolong the lifetime of primary products of dislocation activity and allow to better separate point defect-related from dislocation-related effects. Tarbaev et al. (1988) applied very little strain (of the order of lo4) to CdS and CdSe crystals at low temperature, and measured optical absorption and luminescence spectra; by use of a scanning electron microscope (SEM) in cathodoluminescence (CL) mode, lumines-

333

cence activity could be attributed directly to slip bands: the activity was found not only around dislocation etch pits but clearly also behind dislocations in their wake. The authors found new peaks at a corresponding wavelength in both absorption and luminescence spectra. They attribute them to optical transitions between electronic states of one particular type of point defect complex. Piezospectroscopic investigation reveals C,symmetry for the electronic states and probably also for the defects. All optical modifications of the crystal caused by plastic deformation disappear by storage of the crystal at room temperature. Following the luminescence (L) activity for higher deformation degrees, it turns out that nonradiative centers must be destroyed by moving dislocations, while the new L centers, mentioned above, are produced. As a result the radiative efficiency of the crystal increases by deformation. The PL left behind by moving dislocations in CdS have been imaged; a domain structure of the polarization direction of the emitted light demonstrates collective behavior of the recombination centers. A model of those centers is outlined (Ossipyan et al., 1987). For CdTe a number of investigations have been carried out to detect possible deep dislocation core states and to determine their position. Thermopower and Hall effect data from Muller (1982) and Haasen et al. (1983) for n- and p-CdTe are interpreted to give evidence for a defect level at E,+(0.3 ... 0.4 eV) in CdTe due to plastic deformation. DLTS measurements at n-CdTe by Gelsdorf and Schroter (1984) revealed a mid-gap line at E,-0.72 eV after plastic deformation. Since the trap density of this line turned out one order of magnitude larger than that of possible dislocation core states, the level was interpreted to be due to deformation-induced point defects. In deformed p-CdTe Zoth (1986) detected a DLTS line at E,+

334

6 Dislocations

0.7 eV with trap density about one tenth of possible dislocation core states which might be associated with the latter. The defect levels observed by Zoth and Gelsdorf recently were confirmed by Nitecki and Labusch (1988) by means of photocapacity investigations, where the logarithmic time dependence of the filling factor was also reconfirmed for both levels. Zoth’s results are in agreement with theoretical investigations (Teichler and Grohlich, 1987) (cornp. Sec. 6.4) which predict core states at Cd (g) partial dislocations with levels up to 0.68 eV below E, (and levels at Te (g) partials up to 0.15 eV above E”).

6.4 Theoretical Investigations about Electronic Levels of Dislocations Many theoretical studies have been carried out in the last twenty years to determine the position of possible deep electronic levels introduced by the dislocations in the band gap of semiconductors. The level position depends strongly on the arrangement of atoms and bonds in the dislocation cores, and therefore the field of level calculations is intimately woven together with the field of core structure simulations. Questions of importance in this context concern the detailed atomic arrangement in the dislocation cores; the deep electron level structure of a given arrangement: the relative stability of different core configurations of the same dislocation; and the formation energy, atomic structure, and electronic level position of local imperfections in otherwise perfect dislocations. The theoretical developments in this field are strongly influenced by the fact that the introduction of dislocations in a lattice changes the topology of the whole system. This prevents one from being able to characterize dislocations simply by local

perturbations in a more or less perfect lattice, which is usually possible with point defects or even surfaces.

6.4.1 Core Structure Calculations Present knowledge regarding the atomic arrangement in dislocation cores primarily comes from computer simulations. The core structure is determined through numerical minimization of the total energy as a function of the atomic positions, making use of suitable models for the structure-dependent part of the energy. For straight, ideal dislocations, the arrangement is periodic along the dislocation line. Therefore, the whole atomic pattern can be described by specifying the atomic positions within a flat cylindrical region around the dislocation (with a thickness of usually one or two atomic layers normal to the line). Regarding the pattern in this orthogonal layer, two different approaches are used. The “isolated dislocation” treatments take advantage of the fact that the atomic positions sufficiently far away from the core are known in terms of the elastic deformation field. Thus, to model the core structure, an ensemble of typically some hundred up to some thousand atoms is embedded in an elastically deformed lattice as the boundary. In order to circumvent the embedding process, the “supercell” treatments consider periodic arrays of parallel dislocations where the periodically repeated cell describes the core structure of the dislocations and their nearest environment. To avoid macroscopic stresses and to make the treatment applicable to partial dislocations, arrays of dislocations with alternate Burgers vectors are considered, making use of cell sizes of some hundred atoms in the plane normal to the dislocation. Here the core structure is studied as evolving under the influence of adjacent dislocations, assuming that the cell size is large enough

6.4 Theoretical Investigations about Electronic Levels of Dislocations

to avoid significant effects on the atomic arrangement. Due to the covalent interatomic bonding, different atomic patterns can be obtained for the core of nominally the same dislocation, predicting different energies and properties for the dislocation. Beyond kinetic hindrances, thermodynamics may govern the probabilities for realization of the different core variants in nature. The early core structure simulations are carried out by modeling the energy of the system within a "valence force field" (VFF) approach. This was of significant influence on the notation used to characterize different core configurations. The valence force fields for the diamond structure rely on the existence of covalent bonds among neighboring atoms with the coordination number of the atoms fixed to four. They model the structural energy of an array of atoms in terms of deformations of these bonds, Le., by bond-stretching and bond-bending contributions. In order to treat atoms with lower coordination numbers, these models use the notation of dangling bonds along with the concept of bond breaking. This concept implies a qualitative difference between "deformed" bonds and "broken" bonds, where deformed bonds suffer strong restoring forces with increasing deformations, while broken bonds exhibit no restoring force at all. A rather important parameter in this context is the bond-breaking energy, i.e., the asymptotic energy value ascribed to a broken bond, which measures whether bond deformation is energetically more favorable than bond breaking. As discussed later, from the present view of the electronic nature of covalent bonds, the socalled dangling bonds in many cases mean weak electronic bonds to adjacent atoms, turning the latter into a fivefold coordination. In the VFF picture, the so-called unreconstructed configurations contain as many dangling bonds as demanded by geometry

335

when extending the elastic far field to the core region. From these variants, configurations of reduced density of dangling bonds are obtained by reconstructing neighboring pairs of dangling bonds under slight displacement of atoms, thereby gaining covalent bond energy on account of an increase in lattice deformation energy. The classification of the core configurations according to the density of dangling bonds has become of importance since the electron-theoretical calculations revealed that for the dislocations studied so far deep electron levels occur only in the case of dangling-bond-carrying cores. The theoretical studies concentrate on the glide-set partial dislocations, which according to weak-beam electron microscopy observations are considered to predominate. For Si and Ge in particular the 30" and the 90" partial dislocations are treated, resulting from dissociation of the perfect 60" and screw dislocations, as well as 60" glide-set partials resulting from dissociation of the complete 90" dislocation. As shown in Fig. 6-16, the reconstruction is accompanied by a doubling of the translation period along the dislocation in the 30" partials. In the 90" partials a breaking of the mirror symmetry normal to the dislocation takes place. Consequently, the different core configurations are distinguishable according to symmetry, in addition to their density of broken bonds. Recently, a different type of core configuration was proposed by Bennett0 et al. (1997) for the reconstructed 90" partial, which is accompanied by a doubling of the translational period. At present, there is an ongoing discussion on the energetic ranking of the two types of reconstruction in silicon (Nunes et al., 1998) and on the influence of dislocation interactions on this ranking (Lehto and Oberg, 1998). The unreconstructed 90" partial is a prominent example of the fact that nominal

336

6 Dislocations

Figure 6-16. Atomic pattern in the glide plane of (a) the reconstructed 90" partial and (b) the reconstructed 30" partial dislocation in Si (dashed: unreconstructed configurations).

dangling bonds may show marked residual interactions with their environment, in this case particularly with the two adjacent dangling bonds across the dislocation core. Accordingly, between the corresponding atoms weak bonds are formed (Teichler, 1989) with properties between true dangling bonds and complete covalent bonds, yielding the core atoms to be considered either as fivefold coordinated with two weak

bonds each, or as threefold coordinated. Duesberry et al. (1 99 1) introduced the notation of "symmetrical reconstruction" for this situation in the core of the nominally unreconstructed 90" partial, while Bigger et al. ( 1992) denoted it as "quasi-fivefold". For the 60" partials, early analysis considered one completely reconstructed configuration, as well as partially reconstructed configurations (Hirsch, 1979; Jones and Marklund, 1980; Veth and Teichler, 1984). Recently, a further completely reconstructed variant with modified atomic pattern in the core (Teichler and Wilder, 1997; Lehto, 1998) was studied. The latter variant turned out to have rather similar total energy to the earlier, but lower core energy, where the core energy gain is widely compensated for by additionally stored deformation energy in the elastic far field. The two variants of the reconstructed 60" partial, as well as the two variants of the reconstructed 90" partial mentioned earlier, are spectacular examples of the need to use highly elaborated theoretical approaches in order to estimate the core energy and the elastic energy in the far field with sufficient precision to judge the energetic ranking of different core variants. For compounds, computations were carried out for unreconstructed 30" and 90" glide-set partials, e.g., in GaAs and CdTe. In the following we shall first concentrate on the elemental semiconductors and postpone the results concerning the compounds to the end of the section.

6.4.2 Deep Electron Levels at Dislocations The theoretical calculations of the deep levels at dislocations extract the levels introduced by the dislocations from the comparison of calculated energy spectra for crystals with dislocations with those of ideal crystals. Such an approach seems indispens-

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able, since, as mentioned above, the topological alterations brought about by dislocations prevents one from treating them in a perturbative way. The atomic positions in the crystal with dislocations are taken from independent studies such as the valence force field simulations described above. No significant effects on the level position were found when using simulations from different valence force fields. In order to model the Hamiltonian, either the tight-binding interpolation scheme was used (Jones, 1977; Marklund, 1979; Jones and Marklund, 1980; Veth andTeichler, 1984; Bigger et al., 1992), or the more advanced LCAO approach (Northrup et al., 1981; Chelikowsky, 1982; Teichler and Marheine, 1987; Wang and Teichler, 1989),or the simpler extended Huckel approximation (Alstrup and Marklund, 1977; Lodge et al., 1984). The geometrical models range from spherical clusters (Jones, 1977) of typically 7001500 atoms containing one dislocation to supercell approximations (Marklund 1979; Northrup et al., 1981; Chelikowsky, 1982; Lodge et al., 1984; Wang and Teichler, 1989), where pairs of dislocations with an alternate Burgers vector embedded in a supercell (with cell size of about 50-200 atoms in the plane normal to the dislocation) are periodically repeated, up to treatments of isolated partials in an infinite crystal (Veth and Teichler, 1984; Teichler and Marheine, 1987). By means of the supercell approach, the overall translational symmetry of a periodic lattice is restored (although with an extremely large translation cell), which permits evaluation of the electronic level structure by application of the usual methods known for lattices. In the other investigations the continued-fraction recursion method is applied to evaluate the level spectrum at the dislocation, as introduced in this field by Jones (1 977) and developed further by Veth and Teichler (1984).

337

Because of the translational symmetry along the dislocation line, the bound states in the cores combine to one-dimensional Bloch wave-like states with the levels split into one-dimensional bands. The details of the theoretical studies reveal that reconstructed core configurations of the 30" glide-set partial in Si have no deep levels in the band gap (Chelikowsky, 1982; Veth and Teichler, 1984). The same holds for the corresponding 90" partial (Veth and Teichler, 1984; Lodge et al., 1989; Bigger et a]., 1992) - up to perhaps shallow levels near the band edges (Chelinowsky and Spence, 1984) - and for these dislocations in Ge (Veth and Teichler, 1984). For the unreconstructed 30" glide-set partials one band of bound states is predicted which covers the whole, or at least large parts, of the gap (Marklund, 1979; Veth and Teichler, 1984). For the unreconstructed 90" partial in Si two bands are deduced (Teichler and Marheine, 1987; Wang and Teichler, 1989), a lower band filled with electrons near the valence band edge E, and an empty upper band reaching up to the conduction band. The two bands of the 90" partial reflect the two dangling bonds per periodicity length in this dislocation. The bands are separated by an energy gap of 0.05 eV width centered around 0.2 eV above E,. (The earlier tightbinding treatments predicted for the 90" partial in Ge and Si two partially filled narrow bands of width 0.5 eV around E,, but this result is due to an underestimation of the mutual interaction between neighboring dangling bonds in the tightbinding scheme.) For the 60" glide-set partial, a 'partially' reconstructed, danglingbond-carrying configuration was found (Jones and Marklund, 1980; Veth and Teichler, 1984) with levels in the band gap in Ge and Si. The reconstructed configuration has no deep levels in the gap (Veth and Teichler, 1984). In addition, a 'weakly' recon-

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338

6 Dislocations

structed configuration with an extremely stretched bond was found to have an empty bound state split off from the conduction band i n Si at about 0.8 eV above E , (Jones and Markund, 1980; Veth and Teichler, 1984), whereas in Ge this state turns into a conduction band resonance (Veth and Teichler, 1984).

6.4.3 Core Bond Reconstruction and Reconstruction Defects The profound difference in the electron spectrum of dangling-bond-carrying and dangling bond-free configurations was realized in an early study by Marklund (1979). This observation immediately initiated attempts to estimate the energy difference between the different types of core configurations, which, within the frame-work of valence force field descriptions, particularly demands proper estimates of the bondbreaking energy. Besides the uncertainty in this value, additional difficulties arise from the fact that different valence force field models predict different lattice deformation energies, although the models yield rather similar atomic structure. Table 6-2 presents the energy difference between reconstructed and unreconstructed configurations, AE=E,,,,,-E,,,, deduced from various valence force fields, where the dangling-bond energy entering Table 6-2, E,,, is half the bond-breaking energy. The scatter of the results reflects the fact that the different valence force models are constructed to simulate effectively different features of deformed systems. Keating's (1 966) model in its original version, along with its anharmonk generalization by Koizumi and Ninomiya (1 978), is adapted to the elastic longwavelength deformations but overestimates the energy of short-wavelength deformations in the dislocation cores. The modified potential (Baraff et al., 1980; Mauger et al.,

Table 6-2. Energy difference A€ = €,nr,c-€rec (per line length bo) between unreconstructed and reconstructed configurations. A E (eV)

Dislocation Si

90" 2 Ed,-1.7 glide-set 2 Ed,- 1.6 partial 2 Edb-0.6 2 E,,-1.7 2 Edb- 1.03 0.45 0.37 1.05 0.88 0.69 0.69 0.84

30" Edb- 0.25 glide-set Ed,-0.17 partial Edb-0.1 €db- 0.23

0.55 0.80 1.37 0.33 60" 2 Ed,- 1.5 glide-set partial'

Structural energy model Ge

2 E&- 1.5

2 Ed,-0.98 -

0.9

-

-

Edb-0.15 -

Ed,-0.22

-

Orig. Keatinga Orig. Keating, Mod. Keating' Lifson-Warshel Bond charge' Tersofff Tersoffg Keating + LCAO~ LCA TB~ TBTE EDIP] Anh. Keatinga Anh. Keatingb Mod. Keating' Bond chargee Tersoff' SWrn TBTE EDIP~

2 E&- 1.4 Anh. Keating

Marklund (1980); Veth and Teichler (1984); Marklund (1981); Lodge et al. (1984); e Trinczek (1990); Heggie and Jones (1987); Duesbery et al. (1991); hTeichler (1989); Bigger et al. (1992); Hansen et al. (1995); Nunes et al. (1996); Justo et at. (1998); Bulatov et al. (1995); " Nunes et al. (1998); The values refer to the difference between 'weakly' and 'completely' reconstructed core of the 60" partial. a

'

1987) describes suitably well the energy of short-wavelength deformations but underestimates the energy in the elastic field. Tersoff's model, like Stillinger and Weber's treatment, has similar difficulties to properly describe simultaneously the energy of short-wavelength and elastic long-wave-


length deformations, but they use different approaches in the approximate inclusion of both contributions. Applicable to both regions of deformations are, e.g., Weber's ( 1 977) bond charge model, the Lifson-Warshe1 potential (Lodge et al., 1984) or the recent anharmonic bond charge approach (Wilder and Teichler, 1997; Teichler and Wilder, 1997; Dornheim andTeichler, 1999) which thus provide the most reliable estimates for the lattice deformation energies. In the VFF picture, the dangling-bond energy, &, has to be deduced from peripheral considerations, mainly from electrontheoretical arguments, since covalent bonding is of electronic nature. For Si the estimates of E& range from the early value of 0.5 eV (Marklund, 1980) as a lower limit up to the more recent values of about 1.75 eV (Lodge et al., 1984) and 2.05 eV (Teichler, 1990), where, however, the latter estimates refer to bond-breaking energy values without residual interactions of the dangling bonds with their environment. The residual interactions depend on the geometry of the atomic structure and have to be determined for each defect separately. For the dangling bonds in the 90" glide-set partials of Si a reduction of about 0.7 eV was recently deduced by use of a quantum-chemical LCAO model (Teichler, 1989a) yielding an effective &, = 1.35 eV. Estimates of E& for Ge are not very frequent in the literature, but from comparison of the cohesive energies, a lower &b in Ge than in Si seems reasonable. In accordance with this assumption, for the bond-breaking energy without residual interactions a value of 1.65 eV was estimated for Ge (Teichler, 1990). In the case of the dangling bonds in the 90" glide-set partials of Ge, this value may be reduced by about 0.5 eV (Teichler, 1989a) through residual interactions. With regard to the lattice deformation energies in Table 6-2 from bond-charge-mod-

339

el and Lifson-Warshel-potential calculations (which, as mentioned, are more reliable than the Keating data), values for effective E,, of the order of 1 eV (or larger) mean that for all partial dislocations considered so far the reconstructed variants are energetically more favorable than the unreconstructed variants. This also is reconfirmed by nearly all of the more recent treatments beyond the VFF picture. On the level of phenomenological approaches, we here have to mention for the 90" partial in silicon the studies by Heggie and Jones (1987), as well as Duesberry et al. (1991), who used Tersoff's approach for this purpose, and Justo et al. (1998), who applied their environment-dependent, interatomic potential description. [But it has to be noted that the Stillinger-Weber treatment of the 90" partial by Duesberry et al. (1991) did not give the reconstructed structure as the stable configuration, a point that perhaps indicates some difficulties with the used potential.] In order to take into account in the core bond reconstruction the quantum mechanical nature of the covalent bonds, Teichler (1989) applied an LCAO approach, Hansen et al. (1995) an effective-medium tight-binding description, and Nunes et al. (1996) the total-energy tight-binding scheme of Kwon et al. (1994). Furthermore, on the first principles level of the local density approximation, Bigger et al. (1992) studied core bond reconstruction in the 90" partials using a super-cell approach with a small 64-atom unit cell, and Jones et al. (1993) applied it to a small hydrogen-terminated atomic cluster with up to 160 atoms. In agreement with the VFF estimates, all these calculations yield a lowering of the total energy of the 90" partial in silicon under reconstruction, as shown in Table 6-2. For the 30" partial, pairwise reconstruction of the core bonds seems energetically even more favorable than for the 90" par-

340

6 Dislocations

tials. Besides VFF calculations, for the 30" partials in silicon there exist estimates of the reconstruction energy by Bulatov et a]. (1 995) from the Stillinger- Weber model and by Justo et al. ( 1 998) from their environment-dependent interatomic potential description. All these calculations predict a significant energy gain for the reconstructed dislocation core. This implies that the cores of straight dislocations in the energetically stable configuration do not provide any deep electron levels in the band gap. The theoretical treatments of the dislocation-induced deep electron levels discussed so far do not explicitly take into account effects of the finite intra-atomic Coulomb integral U . Finite U effects on the dislocation level spectrum have been considered by Usadel and Schroter ( 1978) in a phenomenological model designed to simulate the undissociated 60" shuffle-set dislocation. They found a Hubbard-type level splitting between the electron- and hole-like excitations in the narrow one-dimensional band of dislocation states, and they showed that most of the Hall effect data on deformed Ge could be fitted by assuming that U = 0.3 eV. (But there was also one set of experimental data where U = 3.6 eV.) Successful fitting of the Si data was not possible by this model. Besides this quantitative approach, a qualitative discussion of finite U effects on the excitation spectrum of dislocations was provided by Grazhulis (1979) assuming a Hubbard splitting between electron and hole excitations. Application of these ideas to the above-described quantitative one-electron results for the 30" and 90" glide-set partials reveals that significant effects of finite U may be expected in the case of the unreconstructed 90" partials with their small energy gap between empty and filled electron states. In the case of reconstructed partials the gap between full and empty states seems too large to speculate about finite U effects,

and the strong instability of the unreconstructed 30" partials against reconstruction indicates that these variants may hardly be accessible to experimental verification.

6.4.4 Kinks, Reconstruction Defects, Vacancies, and Impurities in the Dislocation Cores The results presented on core bond reconstruction raise the question of whether dangling-bond-carrying defects on the reconstructed dislocations account for the observed electronic activity of the latter. Possible defects are, e.g., local reconstruction defects (RD's) on the reconstructed partials (as shown in Fig. 6-17) and complexes of the RD's with other imperfections such as vacancies or kinks in the dislocation cores (Hirsch, 1980) sketched in Fig. 6-18, or vacancies and impurities in the cores. As discussed in Sec. 6.3.1, the vacancies as well as the RD's in complexes with vacancies and in vacancy clusters on the 30" partial are considered as candidates to explain the experimental EPR spectra and the observed conversion between different EPR-active centers by formation and dissociation of these complexes (Kisielowski-Kemmerich et al., 1985; Kisielowski-Kemmerich, 1989, 1990).

Figure 6-17. Reconstruction defect of the 30" partial dislocation.


341

Figure 6-18. Atomic pattern in the glide plane of a reconstructed 90" partial (in Si) with reconstruction defect (A) and a complex of kink plus reconstruction defect (B).

Atomistic models of RD's and corresponding complexes can be constructed by computer simulation as in the case of straight dislocations, e.g., by embedding a cluster of atoms containing local defects into a crystal with dislocation or by studying a crystal with dislocations which has a periodic array of defects in the core. By such treatments, detailed investigations were carried out on the formation energy Ef and the migration energy Em for RD's, kinks, and some of their complexes. Table 6-3 presents estimates of Ef and Emfor such defects on the 30" partials; Table 6-4 presents data for defects on the 90" partials, resulting from various structure energy models up to the recent environmental dependent interatomic potential (EDIP) approach by Justo et a]. (1998). The VFF data for the reconstruction defects clearly indicate that a significant or even the dominant -part of Ef comes from the dangling-bond energy, Ed,. In nearly all cases, Ef is larger than E,, since the local defects are accompanied by additional lattice deformation energy, with the exception of the RD's on the 30" glide-set partials, where local breaking of the reconstruction provides an energy gain due to lattice relax-

ation. Tight-binding calculation show the dangling-bond deep levels of these defects to be near E,. The corresponding values are also included in Table 6-3, as obtained for the computer-relaxed structures. Regarding these values one must, however, take into account that changes in the atomic arrangement induce significant changes in level positions (Heggie and Jones, 1983). This was particularly demonstrated recently for the isolated RD on the 30" partials in silicon. From ab initio calculations, Csanyi et a]. (1998) deduced that in this case the nominal dangling bond of the RD couples to a neighboring atom and turns this into a fivefold coordinated center. According to Csiinyi et al. (1998), the unique electronic structure of this novel complex is consistent with the EPR signature of the R center observed in deformed silicon. In addition, there are indications that on the reconstructed 90" glide-set partial in Si the RD level may be shifted midgap (Teichler, 1989a; Heggie et al., 1989) due to interactions of the dangling bond with its environment which are neglected in the tight-binding approach. Present theoretical estimates of the binding energy for vacancies to the cores of 30"

342

6 Dislocations

Table 6-3. Formation (E') and migration energy (E") of local defects in the reconstructed 30" partial dislocations and electron level positions A€D = €,-E, for computer relaxed structures. Defect

~~~~

Reconstruction defect

Kink'

(RK) (RK) (LK) (LK)

Edb-0.15 2.55 0.84 0.81 0.34 0.65 0.82 0.35 0.82 1.24

~

~

0.15

0.17

Ed,-0.14

0.20

Orig. Keating" KP SWb

swc

EDIP~ LDFe 0.74 1.35 0.82 2.1 2.1

swc TBTE

swc

TBTE~ TB

Kink pair

>1.1

Mod. Keatingh

Kink pair plus RD

2.39 3.56

SWb KPb

Kink plus RDJ (RC) IRC) (LC) (LC)

0.79 2.15 1.12 0.8

1.04

swc TBTE'

0.22

swc

TBTE~

a Veth (1983): Duesbery et al. (1991); 'Bulatov et al. (1995); Justo et al. (1998); eCshnyi et al. (1998); 'Nunes et al. (1998); Huang et al. (1995); Heggie and Jones (1987); RK, LK denote different kink structures; RC, LC denote different kink-RD complexes.

and 90" partial dislocations in silicon are displayed in Table 6-5. In the case of germanium, so far only one estimate of 1.5 eV is available (Teichler, 1990) for the binding of vacancies to the core of the 30" partials. Despite the scatter of the data, a clear tendency is visible for vacancy binding to the cores. Regarding the electronic activity of the vacancies in the dislocation cores, the vacancies have to be distinguished as in fourfold and threefold coordination. Threefold coordinated vacancies are formed, e.g., when the vacancies occupy the central sites of isolated RDs on 90" or 30" partials. The latter situation gives rise to singly-occupied, EPR-active deep levels at the vacancies, as considered, e.g., by Kisielowski-Kemme-

rich (1989, 1990) to explain the EPR activity of 30" partials in silicon. However, recent ab initio local density calculations for hydrogen-terminated atomic clusters (Lehto and Oberg, 1997) found for the threefold coordinated vacancy in the 30" partial an EPR line that differs in its symmetry significantly from the experimental observations, especially from the symmetry data for the Si-Y line at the 30" partials. Regarding this, the origin of the Si-Y line seems to be an open question at present. Fourfold coordinated vacancies may be found in the cores of otherwise fully reconstructed dislocations. Concerning their electronic structure, the theoretical predictions are controversial. Marklund and Wang (1995) obtained for this

343


Table 6-4. Formation (Ef) and migration energy (Em)of local defects in the reconstructed 90" partial dislocations and electron level positions AED = ED-E, for computer relaxed structures. Defect

Si

E' (eV)

Ge

Em(eV) AED (eV)

Reconstruction defect

Edb f 0.37 0.37 1.31 0.04 0.41

Kink

0.72 0.52e 0.28e 0.1 0.48e 0.4 1 0.12

noj 3.2' 1.3' 1.8 3.0h 1.17 1.62

Ef (eV)

Em (eV) AED (eV)

0.65 0.50e 0.25e 0.47e 0.38

Model

-0.02

Orig. Keatinga Tersoff TBTE ' EDIP~

noJ

Anh. Keatinga Orig. Keating Mod. Keating LDA bond charge inharm bond chargeh TBTE'

2.85h 1.05

Kink pair

0.37 0.24

Mod. Keating' Tersoff

Kink pair plus RD

1.8

Tersoffb

Kink plus RD

Edb + 0.95 E,,+0.36 Edb+0.26

0.12

Inh. Keatinga Orig. Keatinge bond chargee

'

a Veth (1983); Duesbery et al. (1991); Nunes et al. (1996); Justo et al. (1998); e Trinczek (1990); Heggie and Jones (1987); Oberg et al. (1995); hTeichler and Wilder (1997); 'Nunes et al. (1998);j 'no' means that no levels are found.

Table 6-5. Vacancy binding energy to reconstructed 30" and 90" partial dislocations in silicon. 30" partials

90" partials

E, (eV)

Model

E, (eV)

Model

1.9

LCAO a SWb LDFC

0.4 1.69 2.0 0.6

Tersoffd Tersoff LDFC anh. bond charge'

2.62 0.9

aTeichler(1990); bDuesberyet al. (1991); 'Lehtoand Oberg (1997); Marklund (1989).

case deep levels in the band gap, as commonly expected. Contrary to this, Lehto and Oberg (1997) conclude from their ab initio local density calculations for hydrogen-ter-

minated atomic clusters that residual interactions among the atomic orbitals in the fourfold coordinated vacancy in the dislocation core give rise to a sufficient strong level splitting to clear the band gap from any deep states. In contrast to the above-mentioned situations where dislocations introduce electronic levels in the band gap, there also exist cases where combinations of dislocations and otherwise electronically active centers may result in inactive complexes. According to recent ab initio quantum mechanical density functional calculations for hydrogen-terminated clusters (Heggie et al., 1989, 1991, 1993; Jones et al., 1993; Umerski and Jones, 1993), in the core of reconstructed 90" partials in silicon particular ex-

344

6 Dislocations

amples of this kind are substitutional impurities like phosphorus, arsenic, boron, and nitrogen. These impurities behave rather similarly in the dislocations. They have a binding energy of about 2.3 -3.4 eV to threefold coordinated sites in the center of RDs. By trapping such an impurity at a RD site, the dangling bond level of the RD and the impurity level are eliminated from the band gap. The impurities, in addition, act as a strong, local pinning centers against dislocation motion, as experimentally deduced by Imai and Sumino (1983). From the calculations, the binding energy of the impurities to normal sites in the core of the reconstructed 90" partial turned out as 0.4-0.7 eV. In the case of normal sites, binding of the substitutional impurity implies breaking of a reconstructed bond in the core and passivating one of the so-created dangling bonds. The remaining dangling bond may easily drift away as an RD, eventually capturing another one of these impurities. Thus dislocations tend to getter and to passivate the dopands.

Ge theory predicts electron states down to 0.1 eV below the conduction band and hole states up to about 0.02 eV above E, (Celli et al., 1962; Claesson, 1979) for undissociated dislocations. In Si (Teichler, 1975) and for dissociated dislocations (Winter, 1978; Teichler, 1979) the corresponding levels are even closer to the band edges. Current photoluminescence measurements (cf. Sec. 6.3.3) at deformed Ge samples indeed show activation energies of 0.15 eV and 0.08 eV (Lelikov et al., 1989), which might be interpreted as shallow levels below the conduction band and above the valence band. For plastically deformed Si, microwave conductivity investigations displayed thermal activation energies of 0.07 eV in p- and 0.08 eV in n-type material (Brohl and Alexander, 1989). The origin of these possible levels and their interrelationship with the hitherto studied deformation potential states is however, an open question.

6.4.5 Shallow Dislocation Levels

Regarding compounds, theoretical studies of deep electron levels are carried out for dislocation core states in GaAs and CdTe, where 30" and 90" glide-set partials are investigated. In AB compounds the defectfree unreconstructed partials are characterized by core rows of dangling-bond atoms of either A or B type (yielding a doubling of dislocation configurations compared to the elemental semiconductors). Since in compounds an electron transfer occurs between the cations and anions in the bulk (the direction depending upon whether the ionic or covalent bonding type dominates), rows of heavily charged atoms would appear in the dislocations if there were no alterations of charge distribution in the cores. Accordingly, for dislocations in compounds, one has the additional problem of

So far, we have considered possible electron levels at dislocations and at defects in dislocations with states confined to the dislocation cores. Beyond this, theory predicts shallow levels corresponding to more extended states, either states associated with the stacking fault ribbon between the two dissociated partials or states of electrons and holes trapped i n the elastic deformation field of the dislocations ('deformation potential states'). Stacking fault states were deduced for Si with levels up to 0.1 eV above the valence band edge (Marklund, 1981; Lodge et al., 1989; Lehto, 1997) and thus cannot account for the deep levels observed by experiment. The deformation potential also induces shallow states where for

6.4.6 Deep Dislocation Levels in Compounds


modeling the charge distribution around the dislocations. Reconstruction of the partials in compounds involves the formation of bonds between chemically identical atoms. It alsorequires, in the 90" case, a shear along the dislocation line. From this, Sitch et al. (1 994) anticipated that reconstruction becomes less likely for the 90" than for the 30" partials, and for both types as the ionicity increases. As concluded by these authors, in II-VI materials the dislocation core is probably not (or only weakly) reconstructed, while it may be reconstructed in the III-V compounds. In accordance with this, by use of a density functional approach for hydrogen-terminated clusters, Sitch et al. (1994) demonstrated for GaAs that both 90" apartials (having an arsenic core) and 90" ppartials (with a gallium core) are reconstructed where reconstruction of the arsenic core is less strong than of the gallium core. The latter is apparent from the fact that only weak As-As bonds are formed. Regarding the problem of kink formation and kink migration in GaAs, Oberg et al. (1995) found from the same computational approach a kink formation energy of 0.3 eV and a migration energy of 1.1 eV for the p-type 90" partials, while for the a-partials values of 0.07 eV and 0.7 eV were estimated. In the latter case, the smaller values relate to the weak core bond reconstruction. Consequently, the greater mobility of the a-partials in GaAs seems to be due to the lower degree of reconstruction. In addition, Sitch et al. (1 994) studied the influence of electrically active impurities on the core bond reconstruction. From their cluster calculations, they concluded that the trapping of acceptor pairs in the dislocation cores destroys the reconstruction of 90" p-partials but strengthens them for a-dislocations. Donors have opposite effects. In the study, it is anticipated that these changes of the cores with impurities give rise to the experimen-

345

tally observed dislocation locking effects (Yonenaga and Sumino, 1989) of donors and acceptors. According to the theoretical modeling, there are no mid-gap levels associated with the reconstructed cores in GaAs (Oberg et al., 1995), but there may be hole traps at gallium cores (Sitch et al., 1994). From the experiments (see Sec. 6.3.5), there are some indications for deep dislocation levels in GaAs in the lower part of the gap and mid gap (Wosinski and Figielski, 1989; Farvaque et al., 1989), in agreement with the theoretical results. However, from the calculations, clear statements regarding the dislocation electron level structure do not seem possible because of a marked sensitivity of the deduced structure to the actual cluster geometry (Oberg et al., 1995). Notwithstanding this, Oberg et al. (1995) observed a clear difference in the electronic levels of kinks in their saddle point configuration in 90" a-and p-partials. The saddle point configuration of kinks on the a-partial has an empty level pulled down from the conduction band bottom, which lies just above the valence band top and which is absent in p-partials. This suggests that negatively charged kinks in a-partials should, in contrast to p ones, have greater mobility than positively charged ones. These findings may account for the experimentally observed doping dependence of the effective activation energy for dislocation motion in GaAs (Oberg et al., 1995). For dislocations in CdTe, there are tight binding calculations of the electronic level structure (Oberg, 1981; Teichler and Grohlich, 1987; Marheine, 1989). Following an early study by Jones et ai. (1981) for GaAs, Oberg (1981) in his treatment of unreconstructed partials in CdTe applied a one-electron picture to evaluate the electronic levels, neglecting any effects of charge redistribution. In order to account for these

346

6 Dislocations

effects, an improved method was deduced and applied to CdTe (Teichler and Grohlich, 1987; Marheine. 1989) where the tightbinding scheme is extended beyond the independent-particle picture taking into account Coulomb, exchange, and correlation corrections in a parameterized way. The effective charge on the atoms and the atomic position are determined "self-consistently" by making use of the molecular orbital approach to compounds. As in the case of elemental semiconductors, one band of deep electron levels results for the unreconstructed 30" glide-set partials and two bands for the 90" partials. For neutral dislocations with vanishing band bending, the dislocation bands in 30" and 90" Te partials are situated around E, with a width of about 0.5 eV or smaller, and the center of the band typically 0.1 eV below E,. For Cd cores the self-consistent treatment gives a partially filled band about 0.5 eV below the conduction band. Here, the proper inclusion of the Coulomb effects seems of importance since neglect of these corrections, as in the earlier treatment, predicts for the neutral Cd partials levels within the conduction bands which would lead to heavily charged dislocations in contrast to the underlying model assumptions of uncharged objects. According to further studies by Grohlich (1 987) and Marheine (1 989), the defect-free Te dislocations are unstable against electron capture. In intrinsic CdTe stable Te dislocations carry a nominal charge of about 0.05 electrons per dangling bond (reduced according to the dielectric constant of 7 in CdTe) surrounded by a positive screening cloud (Marheine, 1989). From experiments, the deep levels in CdTe found after plastic deformation (cf. Sec. 6.3.6) seem to be due to deformation-induced defects surrounding the dislocations. This seems in agreement with the theoretical results for the un-

reconstructed tellurium partials. The deep levels predicted by theory for the unreconstructed cadmium partials demand further studies, in particular concerning the question whether cadmium partials are, perhaps, weakly reconstructed.

6.5 Dislocation Motion 6.5.1 General

Notwithstanding the common glide geometry of diamond-like crystals and f.c.c. metals there is a fundamental difference in the mobility of dislocations in the two classes of materials. Whereas dislocations can move at the temperature of liquid helium in copper, temperatures of roughly half the (absolute) melting temperature are required to move dislocations over noticeable distances in elemental semiconductors. The reason for this is the localized and directed nature of covalent bonding. The disturbance of the lattice caused by the dislocation is concentrated into a much narrower range of the dislocation core, and this produces a pronounced variation of the core energy when the dislocation moves from one site in the lattice to the next one. There is a saddle point configuration in which the core energy is higher by a certain amount (the so-called Peierls potential) than in the equilibrium positions. This holds for partial dislocations as well as for perfect dislocations, but the Peierls potential should be smaller for partials. This periodic profile of an important part of the dislocation energy depends on the direction of the dislocation in the glide plane; the energy minima are deepest for dislocations which are parallel to one of the three (1 10) lattice rows in a (111) plane. This can be demonstrated by deforming a crystal at a relatively low temperature (0.45 T,) and with a high shear stress (Fig. 6-19). The high

6.5 Dislocation Motion

Figure 6-19. p-type silicon TEM: primary glide plane of a two-step deformed crystal (750°C, t = 12 MPa; 320°C, t = 296 MPa). N: so-called noses (extraordinarily wide dissociation).

stress allows the dislocations to turn with sharp bends from one direction to another. As Fig. 6-19 shows, dislocations follow (1 10) energy minima as far as possible. For this reason, 60" and screw dislocations are the basic types in these crystals and 30" and 90" partials, constituting these two types, are objects of many theoretical investigations (Sec. 6.4). At higher deformation temperatures and, therefore, smaller shear stress edge dislocations are prominent, consisting of two 60" partials. Examining lattice models, one notices that the core of a 60" partial is made from alternating elements of 30" and 90" partials (Hirsch, 1979). Applying

347

high shear stress to a silicon crystal containing segments of edge character pinned by two constrictions (Sec. 6.2) transforms within a short period of time the edge dislocations into a triangle of two 60" dislocations. This happens at temperatures as low as 370"C, where 60" and screw dislocations are about 50 times slower. This seems to confirm that edge dislocations can be considered as a dense array of kinks. In what follows, we will summarize first the results of measurements of the velocity of perfect 60" and screw dislocations as revealed by various experimental methods (etch pitting at the crystal surface, X-ray live-topography, cathodo-luminescence). On account of space, we have to discard many details. There are several up-to-date reviews on the subject (Louchet and George, 1983; Alexander, 1986; George and Rabier, 1987).In the second part of this section, dissociation of perfect dislocations into partials is taken into account; it will be shown that 30" and 90" partials not only have different mobilities, but their position in front of or behind the stacking fault ribbon also influences the friction force to be overcome when moving. Those differences can explain some peculiarities of dislocation mobility under high stress. The authors believe that electron microscopic analysis of the morphology of single dislocations frozen in the state of motion will answer some of the questions still open after 30 years of investigating dislocation motion in semiconductors.

6.5.2 Measurements of the Velocity of Perfect Dislocations in Elemental Semiconductors Since the first measurements of dislocation velocities in various semiconductors by Pate1 and co-workers (Chaudhuri et al., 1962), most authors use expressions such as

348

6 Dislocations

Eq. (6-16) to represent their results:

v=uo (&)mexp(--f)

(6-16)

where vo, m, and Q are quantities which first depend on the material under investigation, and second on the type of dislocation which moves. For small stresses ( r c 10 MPa) Eq. (6- 16) is not appropriate because there m depends not only on temperature but also on stress (George and Rabier, 1987). Actually, Q and m are found not to be independent constants, but rather Q increases with a lowering of the (shear) stress, indicating that an important portion of the stress dependence of dislocation velocity is due to a contribution by the applied stress to overcoming the activation barrier (Alexander et al., 1987). From an experimental point of view, it should be mentioned that there is a controversy about the reliability of deducing the dislocation velocity from etch pitting the points where the dislocation penetrates the crystal surface before and after displacement. It is true that the dislocation half loops produced by scratching or hardness indentation show some irregularities near the surface (George and Champier, 1980: Kiisters and Alexander, 1983). Thus it is absolutely necessary to use only the straight segments below those irregularities for measurements. Also, some pinning of dislocations can be inferred from the occurrence of a starting stress, i.e., a minimum stress fordislocation motion. Repeated loading of the same specimen must be avoided in these experiments. On the other hand, X-ray live tropography suffers from the possibility that recombination of electron hole pairs produced by irradiation influences the dislocation mobility (see below for the photoplastic effect). Using the double-etch-pitting technique with sufficient care yields very re-

producible results from laboratory to laboratory at least in the range of lower temperComparison of measureature ( ~ 0 . T,). 6 ments of the dislocation velocity in the same material (n-type silicon) made by etch pitting and X-ray in situ topography showed agreement within 30% (George and Michot, 1982). Since the parameters Q and rn extracted from macroscopic deformation tests with the help of the microdynamical theory (Alexander, 1986) are in satisfactory agreement with the values measured by etch pitting, there is little doubt that the latter technique gives information reflecting properties of dislocations in the bulk. However, this good correlation between macroscopic and microscopic sets (Q, m) holds only for elemental semiconductors (Si and Ge). In compounds, the large difference between different dislocation types obscures the connection between the activation enthalpy Q’ of the state of optimal plasticity (lower yield stress or maximum creep rate) and the Q values of the dislocation movement. Lower temperatures are to be preferred for measuring dislocation velocities not only because of slower gettering of impurities to dislocations but also for theoretical reasons: Although some authors doubt whether the Peierls potential is rate controlling for plasticity at high temperatures, it certainly is rate controlling below 0.6 T,. Thus a comparison with the theory outlined in Sec. 6.6 should be made here. For the “best values” of the parameters Q and m of Eq. (6-16) the most extensive data can be found for 60” dislocations in undoped FZ silicon. Critical discussion (Alexander et al., 1987) reveals Q to be weakly stress dependent in the region 4 MPa 5 t l 200 MPa corresponding to (George et al., 1972)*:

* At r = 300 MPa Q is 1.8 eV (Kusters and Alexander, 1983) instead of 1.95 eV from Eq. (6-17).


[

J

= 2.6 eV -0.1 15 eV In M;a (6-17)

It is satisfactory that activation analysis applied to the yield stress of FZ silicon results in a barrier height AGO=2.6 eV (Omri et al., 1987) and in an activation volume V = -aGlat = E , l t with E , =0.375 eV (Castaing et al., 1981). The stress dependence of the activation volume deduced from Eq. (6-17) is of the same type, but the absolute value is three times smaller. Comparing Eq. (6-17) with the empirical ansatz Eq. (6-16) allows the identification of Q with Qo and m with E,IkT. In fact KisielowskiKemmerich (Kisielowski-Kemmerich, 1982; Alexander et al., 1983) extending measurement of the dislocation velocity in FZ Si to lower temperatures and higher stresses came to the conclusion that the stress exponent m consists of two independent parts: rn = m,

+ m,

(6-18)

where m l =E,lkT with Ei(screw) = 0.092 eV, and Ei(60" dislocations) = 0.122 eV (90I30) and 0.13 eV (30190). The other (smaller) component m, is negative for compression tests and depends in a complicated manner on the deformation geometry (see below). As mentioned above, it is not self-evident that the rate-controlling mechanism for dislocation motion stays the same in the entire temperature range up to the melting point. In fact Farber and co-workers found a sudden change of Q at about 0.75 T, both in silicon and germanium (both Czochralskigrown) (Farber and Nikitenko, 1982; Farber et al., 1981). In silicon Q increases from 2.2 eV to 4 eV, nevertheless, because of a change of the prefactor, dislocations at higher temperatures are more mobile than is extrapolated from low temperatures. For

349

germanium the situation is more complex: in the low-temperature regime ( t:). The authors explain these findings (for Si) in the frame of the HirthLothe diffusion model (Sec. 6.6). The stress (7 MPa) should be so low that the kink density stays at its thermal equilibrium C k = (2/b) exp ( - U k / k T ) ( U k formation energy of a single kink). Under the stress t,kink diffusion provides a drift velocity of kinks V k = D k t b h/kT. ( D k = vD exp (- W,/kT) is the kink diffusivity, h the height of a kink.) The dislocation velocity v under those conditions is v = C k vk h. In the model of Nikitenko et al. (1 985), tT marks the time in which a kink pair grows by diffusion to its critical length; on the other hand, in the time tp*, a kink pair vanishes by back diffusion which during the pulse length ti has reached a length 2 v k ri. From the equation = + 2 uk ( t i - t ~ ) ,Dk can be c a ~ c u ~ a t ed and from that W, = 1.58 eV (for z = 7 MPa). After its publication, this simple explanation of the experimental results was criticized by the authors themselves (Nikitenko eta]., 1987, 1989) and by Maeda (1988). First of all, it cannot be understood why kink pairs should fully shrink by diffusion because the attraction between the two kinks is of short-range type. Second, as shown by simulations, the critical pulse length r? is expected to be much shorter. Maeda ( 1988) repeated the experiments using CY dislocations in GaAs and found qual-

v m

v m

351

itatively the same behavior as Nikitenko et al. Although there were some indications that the problem could be due to experimental imperfections, Nikitenko et al. ( 1 987, 1989) were of the opinion that the simple model does not apply. They maintained that the deviation of the experimental results from the diffusion model is ascribable to point defects which change the energy of a kink pair and also produce the starting stress for dislocation motion. Therefore, they leave during the pauses a stress equal to the starting stress compensating for an effective back stress on the kinks, which can be explained by applying the theory of random forces to kink motion (Petukhov, 1988). In fact, now the shrinkage of kink pairs corresponds better to theory. The authors claim that the modification of the model does not change significantly the estimate of W, (Farber, private communication). A comparison of starting stress and pulse deformation reveals that three quarters of W , (1.2 eV) can be ascribed to the secondary Peierls potential, and one quarter to the effect of impurity atoms. Maeda and Yamashita (1989) on the other hand extended the pulse deformation tests to low-doped (8.6 x 10l6 cm-3 Si) GaAs. The authors found discrepancies between the diffusion model and experiment to be even more serious than with higher-doped crystals: now the critical pulse length r? is more than ten times longer than the time for moving one Peierls distance during static loading! Maeda et al. came to the conclusion that only locking of the dislocation by point obstacles and bowing out of dislocation segments can explain the particular type of intermittent loading characteristics they found. This would mean that the mere existence of an intermittent loading effect is no proof in favor of or against the kink diffusion model. In summary, no clear-cut conclusion can be drawn at this time from the

352

6 Dislocations

intermittent loading effect revealed by the pulse loading experiments. 3) Jendrich and Haasen ( 1988) extended the analysis of internal friction of deformed germanium crystals to higher temperature. They found two peaks with activation enthalpies. H , = 1.1 1 eV and H , = 2.07 eV. Within the framework of the kink diffusion theory and kink-kink annihilation (Sec. 6.6), the authors ascribe H I to the migration of kinks (consuming 70% of the activation energy Q of dislocation motion) and H 2 to the formation of kink pairs.

ing in front (90l30 dislocations). Only 30l90 dislocations are narrowed as expected (the same is true for germanium) (Alexander et al., 1980). Wessel and Alexander (1977) explained those findings by introducing a lattice friction force Ri into the consideration leading to Eq. (6- 19). Then, depending on the actual difference ( R 2 - R I ) ,widening as well as narrowing is possible:

(6-19')

with

6.5.4 Experiments on the Mobility of Partial Dislocations

S =R qd- R- -- RZ-RI - 1 - R "

Application of an uniaxial compression stress (7 to a crystal in general induces different glide forces F, to the two partials of a perfect dislocation (i = 1, 2 numbers, the partial moving in front of and behind the stacking fault, respectively). For compression parallel to (2 13), F , = 0.71 F 2 . Assuming that during steady state motion (v,= v,) the effective force (made up from F , , the attraction yby the stacking fault and the repulsion Ald) is equal for both partials *, one can calculate a stress-dependent dissociation width (Wessel and Alexander, 1977): d=

do

l+f-

bt

(6-19)

27 withfb t = ( F 2 - F , ) . Therefore, freezing in this high-stress state would lead one to expect the dislocations to be narrowed (d < do).But in fact the opposite is true (on average over a considerable scatter) for screws and those 60" dislocations which have the 90" partial mov-

bt

R?+RI

1+R*

In this sense, the ratio R* =R,lR, of the friction stresses felt by the two partials has become a measurable quantity. A large number of measurements could not reduce the wide scatter of the measured width d; this scatter reveals the coexistence of dislocations in the whole range of R* between zero and infinity. This fact and the mean values of R* have been discussed by Alexander (1984). We will repeat here only the conclusions (for T = 420°C). (1) It is not surprising to find the mobilities pi = (Ri)-' of 30" and 90" partials to be different in view of their different core structures (Sec. 6.2). Generally, 90" partials are more mobile than 30" partials. (2) Statement 1 cannot explain that most screw dislocations consisting of two 30" partials are widened. In fact, the position of a partial before or behind the stacking fault also influences the mobility. For that reason, R" (90130) is not reciprocal to R* (30l90). (3) Examining whether R* is a unique property of a given dislocation type W, WeiB (1980) measured R* for five different glide systems. It turned out that the order of magnitude in fact stays constant (RfCrew= 0.5,


R50/90 = 3, R$0,30= 0.35), but the differences from one glide system to the other were well outside of the scatter of one system. Grosbras et al. (1984) made similar experiments and proposed that the climb force acting on each partial - including its sign should be responsible for the difference. Heister (1982) (cf. Alexander, 1984) tried to find the link between the velocity of aperfect dislocation and the friction forces acting on its partials both measured at 420°C and at high stress. He succeeded by assuming that the stress exponent m is the same for the partials as well as the total dislocation. Calling v0 exp (-QIkT) in Eq. (6-16) voT,he writes *: 1 -

rb=Rl+R2=(&:mMPab

(6-20)

Using for each partial a similar ansatz 1

the left and the right side of Eq. (6-20) fit together only if m,=m2=m Thus

[&Im

I

\v02

1 -

=

MPa b

(6-21)

The ratio of friction forces then is 1 I

R2

R02 b o 1

\ -

I

* MPa is the dimensionality constant from Eq. (6-16).

353

Using the exponents m as measured under the same conditions (Kisielowski-Kemmerich, 1982) as the R*, one can separate ~ , ) 6-6). With (Ro,/Ro2) from ( ~ ~ ~ / 2 /(Table these 6 numbers, Heister could fit 14 out of 15 measured dissociations d (3 dislocation types in 5 systems). Moreover, the four parameters Rol, RO2, vel, and vO2can be separated by putting absolute values of v into Eq. (6-21). Since for 30/90 dislocations velocities were measured in four systems, two systems provide control of the rather good fit. Table 6-7 presents the best values obtained for the four parameters of the three basic dislocation types in silicon. Calculating the combination ROiI(~oi)l'm being characteristic for the (inverse) mobility of the related partial (Table 6-7), the result is fairly interesting: there is satisfactory agreement for the two 60" dislocations: the 90" partial is about three times more mobile than the 30" partial. However, the 30" partials of screws are clearly less mobile than the 30" partials belonging to 60" dislocations and, in addition, the trailing partial is appreciably less mobile than the leading partial. Closer inspection of Table 6-7 reveals the stress exponent m to be mainly responsible for the difference between 30" partials both in screw dislocations and in 60" dislocations. For comparing different glide systems in Table 6-7, m is approximated by its geometry-independent part m, = Ej/kT.The smaller mobility of 30" partials in screws, therefore, appears to correspond to E , (screw) e E , (60" dislocation). The special character of 30" partials out of screws is particularly interesting due to the fact that the dislocation related EPR active centers seem to be located in just these 30" partials (Sec. 6.3.1). As shown by Rabier and Boivin (1990), the idea of a change in the dissociation width of screw dislocations under high shear stress

354

6 Dislocations

is the key to understanding the plastic properties of GaAs (and probably other compounds as well) in the low temperature range (from 200°C down to room temperature). Irrespective of whether a-or p-30" partial dislocations are less mobile, either positive or negative screw dislocations will be narrowed because each screw is composed from an a- and a b 3 0 " partial. Actually, in the temperature range just mentioned, stress-strain curves exhibit some peculiaries when compared with medium temperatures (Rabier, 1989): (1) The yield point phenomenon is absent, and the curve is parabolic. This also holds for GaSb and InP. (2)The doping influence of the elastic limit is reversed. While at medium temperatures the yield stress of n-type GaAs is about five times larger than that of intrinsic and p-type GaAs, at room temperature for n-type GaAs it is smaller than that for the other two materials by about a factor of two. A similar trend has been found for InP. The authors (Rabier and Boivin, 1990) give convincing evidence that plastic deformation in the considered temperature (and stress) range is controlled by cross slip of screw dislocations. This is supported by TEM investigations. The threshold stress t, for cross slip can be calculated from Escaig's theory ( 1968), which assumes constrictions to be present and to provide nuclei for dissociation on the cross slip plane. The activation energy We for cross slip then depends on the ratio dld' of the dissociation widths on cross slip plane and primary glide plane, respectively: the larger the rating dld', the smaller the value of W,. For the usual compression axis of the type (213), the cross slip plane is parallel to the compression axis, so that d is equal to the equilibrium width do. From Eq. 6-19 it follows: d

-

d'

2y

with R* = b / p , .Quantitative calculation for GaAs (Rabier and Boivin, 1990) showed that W , at small stress (10 MPa) amounts to 6 eV and is practically independent from R*. However, when t approaches the high stress regime, W , decreases reaching a certain value (e.g., 1.5 eV) at a lower stress for a higher value of R*. (Large R* means d e d o , ) Therefore, cross slip is prominent for n-type GaAs at a lower stress than for intrinsic and p-type GaAs in agreement with what has been observed (item 2 above). The particular shape of the stress-strain curve (item 1) also is explained by the decisive role of cross slip of screws. At the beginning of deformation, surface sources produce dislocation systems consisting of long parallel screw dipoles extended by fastmoving 60" dislocations of a type (Kesteloot, 1981). Several such loops emanate from each source, and the back stress from those loops eventually blocks the source, so that steep strain hardening results. But when the stress reaches a level where stress-induced narrowing of screws becomes important, cross-slip of screws out of the slip plane relaxes the pile-ups and the back stress on the sources. At the same time, spreading of glide onto new glide planes by double cross slip provides new dislocation sources. Both processes continuously reduce the slope of the stress-strain curve: dynamical recovery causes the deformation curve to take a parabolic shape. Different behavior of the two types of 60" dislocations is observed also with respect to the phoroplastic effect (PPE) (Kiisters and Alexander, 1983). Here the 30/90 dislocations show strongly increased mobility when illuminated with bandgap light, while the 90/30 dislocations show little or no effect. This distinguishes the PPE from doping influence, which always concerns both types of 60" dislocations equally, and which may be superimposed to the PPE. The PPE


can be rationalized as a decrease of the activation enthalpy Q by 0.68 eV (Si). The related effect was found by Maeda and Takeuchi ( I 98 1, 1985) in A1"BV compounds and was called radiation-enhanced dislocation glide (REDG). Both grops interpret the enhancement of dislocation mobility by light (or irradiation with electrons) as recombination-enhanced kink motion. A different model is proposed by Belgavskii et al. (1985). The effect of electrical doping (Alexander, 1986) and of alloying (Sumino, 1989) on dislocation velocities has repeatedly been reviewed. Theoretical models are discussed in Sec. 6.6.2. In view of limited space, we refer to those references. Achievement of a better understanding of the interaction of oxygen atoms with moving dislocations in CZ-silicon by the work of Maroudas and Brown (1991) has to be mentioned. The authors calculated numerically the distribution of oxygen around moving 60" dislocations in the temperature range 600-1500 K for various dislocation velocities. The steady state is treated for the dislocation core being saturated. But the moving dislocation continuously has to dissociate from individual oxygen atoms which are replaced by new atoms. Therefore three components contribute to the drag stress to be overcome by the dislocation: the pure lattice (Eq. 6-16), the interaction with the oxygen cloud around the dislocation and the force necessary to dissociate from the oxygen in the core. The resulting dependence of the dislocation velocity v on shear stress and temperature agrees well with the experimental results of Imai and Sumino (1983). Recently, the existence and some properties of kinks in partial dislocations in silicon were demonstrated by HREM (high resolution electron microscopy) (Alexander et al., 1986; Kolar et al., 1996; Spence et al., 1997; Alexander et al., 1999). For this project, plane view orientation of the specimen

355

(electron beam perpendicular to the glide plane) had to be chosen. In this orientation, the projection of the lattice around dislocations into the image plane cannot be interpreted in a straightforward manner. Fortunately, the stacking fault ribbon between the two partials directs intensity into "forbidden" reflections of the type 1/3 (4223. Using these reflections for imaging (resolution 0.3 nm), partials may be localized with sufficient accuracy to detect kinks and to estimate their density. Moreover, the change of the dissociation width d can be measured when the dislocation from nonequilibrium width d relaxes to the equilibrium width do (this is an HR version of the method described in Sec. 6.5.3). First, a high kink density of the order of 4 x 1O8 m-' is observed (in the 90" partial); this is surprising as dislocations introduced by high-stress deformation which look straight on a weak beam scale (1 -1.5 nm resolution) are being investigated. It is interesting to note that the kink density in 30" partials turns out to be three times smaller. This is in agreement with the ratio of mobilities of the two basic types of partials in silicon (see Table 6-4). Relaxation experiments (by heating in the microscope) were undertaken with 60" dislocations of both types (30/90 and 90/30). The authors could notice motion of the 90" partials only. In one series of experiments they restricted the annealing temperature to 130 "C, assuming that under these conditions the formation of kink pairs can be neglected. To reduce radiation damage, the beam was turned off during relaxation. From the distance the 90" partial moved within a certain time and the kink density, the kink velocity vk (- 2 x m s-l) can be determined. The shear stress t acting on both partials could be calculated from the distance d. Because it was not possible to repeat such measurements over a sufficient temperature range,

-

356

6 Dislocations

the activation enthalpy of (partial) dislocation motion W , had to be calculated on the basis of the kink diffusion model by Hirth and Lothe (Eq. 6-23). It turned out as W , = 1.24 2 0.07 eV ( T = 130"C, t = 275 MPa)

A somewhat different relaxation experiment was started at 600°C. In this case, dislocation motion was documented on video film, and the electron beam had to be working all the time. The 90" partial moves by the collapse of kink pairs. Subtracting video frames before and after the event minimizes image noise due to surface roughness. Here the activation enthalpy W , appeared considerably higher W , = 1.7 f 0.1 eV (600"C, r = 108 MPa)

We believe that the value 1.24 eV represents the Peierls potential of the second kind. It is in reasonable agreement with earlier numbers determined by weak beam electron microscopy (Hirsch et al., 1981: Louchet, 1981; Gottschalket al., 1987). Most recently, Teichler and Wilder (1 997) obtained W,= 1.17 eV using a new inharmonic bond charge model. The difference between the activation enthalpies measured under different conditions remains a challenge for further work. Influence of the beam in the second case should decrease W , (see Sec. 6.5.4). Considering the different acting stresses, an activation volume V, may be introduced. The simple approach, U , = W,- V , t results in V,=0.44 nm', but the activation volume of dislocation motion seems not to be a constant (Sec. 6.5.2). Evaluation of the formation enthalpy Fk of kinks from these experiments is a much more difficult problem, because the distribution of kinks in the dislocations has been formed during the last deformation step under high stress (-250 MPa). So the high

stress theory by Hirth and Lothe (1982) has to apply. This has been tried using limited material; the estimate is Fk 0.73 f 0.15 eV (90" partial, 420 "C, 250 MPa)

Following the Hirth-Lothe theory of kink diffusion in the kink annihilation case, the activation energy Q of dislocation motion is expected to be the sum of F , plus W , (for a more exact deduction, see Sec. 6.6.1). The HREM result (Fk+ W,= 1.97kO.22 eV) is to be compared with the compilation of the mass of macroscopical measurements (of perfecr 60" dislocations) given in Eq. (6- 17); it shows: Q (250 MPa) = 1.96 eV. Even if it is accidental that the agreement appears perfect, the authors hold the view that it shows that the new technique is able to bring to light the properties of kinks and the motion of partials on a nearly atomic scale. It is interesting to note that the activation parameters are not very different, whether considering perfect or partial dislocations; in fact, the different mobilities of 30" and 90" partials (Sec. 6.5.4) reflect only small differences of the activation enthalpies. Concluding this paragraph, it should be noted that at 600 "Cinterruption of the motion of 90" partials for several seconds has been observed, from which it can be concluded that obstacles had to be thermally overcome, the activation energy being 2.4 f 0.04 eV. [The lacking of such obstacles under low-dose conditions (at 130°C) is explained by recombination-enhanced diffusion of impurities under the beam.] Ab initio cluster calculations by Umerski and Jones (1 993) point to pairs of oxygen atoms bonded to the reconstructed dislocation core (bonding energy 2.5 eV). In summary, HREM supports the ideas put forward by Hirth and Lothe (1982) about dislocation motion under a high Peierls po-


tential. However, serious problems remain; above all, the discrepancy between the large formation enthalpy Fk of kinks and the high kink density along partials. We will not speculate about explanations, as new theoretical activity brings to light (mostly by computer simulation) a much more complicated spectrum of core defects in partials in silicon than considered hitherto (Duesbery and Richardson, 1991; Duesbery et al., 1991; Bulatov et al., 1995; Nunes et al., 1996; Bulatov et al., 1997). With the tools of molecular dynamics, it is now possible to check the low energy paths associated with the nucleation and motion of defects (Bulatov et al., 1995). Here we can only specify some of the old and new ideas that are of prime concern: 1) The atomic structure of kinks does not just depend on the character of the (partial) dislocation they belong to. If a kink pair is considered, it is necessary to distinguish between the left and the right kink. Depending on several symmetry arguments, the two may be of different or equivalent structure, formation enthalpy, and mobility. In particular, the kinks must be different when the dislocation is of mixed character (30” partial! j (Bulatov et al., 1995). Nunes et al. (1996) found a second reconstruction mode of the 90” partial, which in contrast to the mode described in Sec. 6.4.3 doubles the period along the partial. The possible coexistence of segments of the two modes and the conception of “half kinks” make things extremely complicated. 2) Kink-soliton complexes seem to be important elements of motion; solitons running along the partial transform left into right kinks and vice versa. 3) Vacancies seem to be enriched in the core of certain partials (EPR results); if so, they will form complexes with other core defects.

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4) Besides the glide segments, some shuffle segments may also exist along a partial (Blanc, 1975). They may be more mobile than the first named (Louchet and Thibault-Dessaux, 1987). Equilibrium between both types is effected by vacancies (Alexander, 1974). 5 ) Point defects moving with the dislocation can influence its motion in different ways (Iunin and Nikitenko, 1999). 6) Point defects and impurities left behind by moving dislocations change the lattice resistance for subsequent dislocations (“extrinsic effects”) (Farber and Iunin, 1985). 7) All components of the stress tensor working on the dislocation core have to be taken into account. The “computer experiments” mentioned earlier open a new area of dislocation research. Possibly the remarkable scatter of careful experimental results, which had to be reported in the section on measurements of dislocation velocity and splitting width, will be explained by the different reactions by which dislocations can move. On the other hand, a rather simple function exists describing average dislocation motion [Eqs. (6-16) and (6-17)], which has to be met by the simulations. 6.5.5 Compounds

Dislocation velocities in compounds are mostly represented by relations of the type of Eq. 6-16. At equivalent temperatures (with respect to the absolute melting temperature Tm),dislocations are more mobile in A”’BV compounds than in elemental semiconductors. This fact is a source of problems for crystal growers. Because in compounds, a and dislocations are distinguished by quite different values of the parameters v0, m, and Q of Eq. 6-16, the ac-

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6 Dislocations

tivation energy of macroscopic plastic deformation (i.e., of the lower yield stress or maximum creep rate), is not simply related to the activation energies Qcrp of dislocation motion. In comparing a and p dislocations, one generally finds that a dislocations (with anions in the core when belonging to the glide set B ( g ) ) are more mobile than /3 dislocations, but this differnce strongly depends on doping: in p-type GaAs it may be even reversed. By high stress deformation, the order of mobilities of partial dislocations can be determined, as described above. It turns out that in undoped GaAs the mobility increases in the ordery(30P) < y ( 3 0 a ) < p ( 9 0 a p ) , while in p-type GaAs the order of the two 30" partials is reversed (Androussi et al., 1987). The influence of irradiation on the mobility of dislocations in AI11Bv compounds has been mentioned above. The literature on dislocation motion in GaAs has recently been reviewed (Alexander and Gottschalk, 1989). Turning to A"BV' compounds, Lu and Cockayne ( 1 983) showed that also in those crystals dislocations stay dissociated when moving. Unfortunately, measurements of the velocity of single dislocations by classical methods are impossible because stationary dislocations become pinned (Ossipyan et al., 1986). Thus information on dislocation motion is gained from the electric current carried by groups of dislocation during cyclic deformation. In fact, the plastic strain rate is measured, and activation analysis is applied to ( T , t).It is not clear if, and in which temperature range, a Peierls mechanism is active. Clearly, the electric charge carried by the dislocation is a determinant for the dislocation mobility. There are several mechanisms conceivable for this: The electrostatic energy of a kink in a charge line (Haasen, 1975), interaction of a charged dislocation with lattice rows of ions and/or

with charged point defects (Ossipyan et ai., 1986). A"BV' crystals are plastically deformable down to much lower temperatures than other semiconductors and they exhibit a large positive photoplastic effect (PPE), i.e., during illumination the flow stress of a crystal drastically increases. This effect by most authors is interpreted as indicating reduced mobility of dislocations, but Takeuchi et al. (1983) claim to have proof of a reduced mean free path of dislocations. In CdS and CdTe there is also a negative PPE as in A111Bv compounds (Ossipyan and Shiksaidov, 1974; Gutmanas et a]., 1979). In summary, dislocation motion in A"BV1 compounds shows many interesting properties, but it is far from being understood, even in the limited sense reached for elemental and A"'BV semiconductors.

6.6 Theory of Dislocation Motion 6.6.1 Dislocation Motion in Undoped Material A number of theoretical models for dislocation dynamics have been derived to account for the experimental observations about the dislocation velocity discussed in the preceeding section. The development of these models is an interesting example of the interplay between theory and experiment, where the theory at present, however, has not reached its final form. There are some common features in the theories of dislocation dynamics which are based on rather general assumptions and there is a large number of facets which have changed with time but nevertheless provide the framework of the current discussions. We will give a descriptive presentation of the actual state of the field by avoiding as far as possible detailed mathematical derivations. A

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6.6 Theory of Dislocation Motion

review concerning the quantitative comparison between theory and experiment and including most of the necessary formulae was recently provided, e.g., by Alexander (1986). Three approaches to the theory of dislocation motion in undoped material will be presented here: the kink-diffusion model, as described by Hirth and Lothe (1982), the so-called ‘weak-obstacle’ theory of Celli, Kabler, Ninomiya, and Thomson (1963), and a modification of the weak-obstacle theory proposed by Moller (1978). The latter theory accounts for the fact that dislocations in semiconductors dissociate into partials, whereas the first two theories mentioned consider complete dislocations. In addition, theoretical approaches about the dislocation mobility in doped material will be discussed. The theories of dislocation motion make use of the fact that in semiconductors the dislocations induced by deformations (except at high temperatures) are straight and aligned along a (110) direction of their { 1 1 1 ] slip plane, as revealed by Lang topographs and transmission electron microscopy observations. This behavior indicates a pronounced Peierls potential with Peierls valleys parallel to the (1 10) lattice rows. The dislocation velocity measurements are carried out with these straight screw and 60” dislocations (and their partials) forming hexagonal loops in the glide plane. The motion of such an “ideal” dislocation proceeds by kink propagation. The high Peierls potential (compared with the dislocation line energy) induces rather narrow kinks with kink width of the order of half the Burgers vector length b ( b measures the fundamental translational period of the atomic arrangement along the dislocation line). Such a small kink width means that the dislocations passes from one Peierls valley to the next within a line segment of length b as indicated by the atomistic model in Fig. 6-1 8.

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Both features, the high Peierls barrier and the small kink width, are a consequence of the covalence bonding in the system. The fundamental elements of the theory of dislocation motion in this picture are the thermal nucleation of kinks, the propagation of the kinks along the dislocations, and, finally, the termination of the kink activity. Regarding the process of nucleation it commonly is accepted that by thermal fluctuations, kink pairs (called “double kinks”, DK’s) are generated with kink-kink distance larger then a critical length s*. s* is governed by the fact that for s >s*, spreading of the DK’s under the action of the local stress t is energetically more favorable than regression, whereas for s c s* the increasing attractive kink interaction favors collapsing of the DK. With nucleation rate J for stable DK’s, the dislocation velocity becomes (6-22)

(vk: kink velocity, f D K : mean lifetime of a DK). Propagation of kinks may proceed as viscous flow or by thermally activated steps, the latter yielding a diffusive motion. Since kink movement in the semiconductors requires breaking and reconstruction of covalent bonds, which demand rather high energies compared to kT, this motion commonly is considered to be thermally assisted, giving for vk under stress t tbh

V k = -U 2 YD

kT

exp (- W, l k T )

(a: diffusion step length,

(6-23)

vD: Debye fre-

quency, W,: activation enthalpy), the relationship already mentioned in Sec. 6.5.3. According to Hirth and Lothe (1982), the nucleation rate can be calculated under the condition of slow kink motion yielding J = ( v k / b 2 exp ) (-EgK/kT)

(6-24)

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6 Dislocations

where the activation energy E g K corresponds to the formation energy for a critical DK of width s* and is slightly stress dependent. The proportionality of J with uk reflects that the uk scales the escape rate of DK’s from the critical region around s*. Within this Hirth-Lothe approach, it seems quite natural to identify W , with the secondary Peierls potential, i.e., with the periodic variation of the kink self-energy along the dislocation due to the periodicity of the atomistic structure. Internal friction measurements on Ge by Jendrich and Haasen (1988) show two damping maxima, occurring after deformation only, with activation enthalpies of (1.108+0.01) eV and (2.07k0.2) eV. With regard to a number of independent internal friction studies after 1972 which all failed to reproduce an earlier 0.1 eV peak, Jendrich and Haasen (1988) came to the conclusion that this peak cannot be attributed to geometrical kink motion, because then it should have been detected by the later investigations, but may have been produced by point defects sensitive to purity and annealing treatment. They attribute their 1.108 eV peak to the movement of geometrical kinks on single partials, since this should be the process with lowest energy, giving W,= 1.108 eV. The second peak is attributed to the formation of kink pairs, yielding EgK=2.07 eV. These conclusions are in accordance with computer simulation by Jones (1985) for Si, who deduced from an atomistic lattice model a value of W,= 1.3 eV for kmks on the reconstructed 90” glide-set partials (1.4 to 1.9 eV for different variants of kinks on the 30” partials) in agreement with an early internal friction measurement by Southgate and Attard ( 1 963). The individual kinks contribute by their movement to the dislocation motion, until the kinks become immobile by reaching some impassable barrier (“strong obsta-

cles”, e.g., a node, a long jog, or possibly a sharp corner of the line) or until they are annihilated with an opposing kink from a neighboring DK. Depending upon whether immobilization (case I) or annihilation (case 11) predominates, the mean active time of a DK, fDK, is determined either by L / 2 uk (with L the mean spreading width of a DK before immobilization, i.e., the average distance between the strong obstacles) or by Lann/uk(with La,,= 1/(J fDK)the mean kink path before annihilation). For these situations u becomes

hJL, (case I) 2 h IIJvk, (case 11)

(6-25)

In case I the velocity u scales with L and is thermally activated with enthalpy Q = EgK+ W,. In case I1 the velocity is independent of L and has an activation enthalpy Q= + W,. u exhibits a lower increase with temperature in case I1 than in case I since the mean free path by which the kinks contribute to u is reduced with increasing density of kinks. According to experiments within the electron microscope (Louchet, 198 1 ; Hirsch et al., 198 l ) , the dislocation velocity in Si is proportional to the length L of the segments as long as L does not exceed some 0.2 pm. This indicates that La,, = 0.2 pm.It implies that under normal conditions (i.e., L 2 0.2 pm) kink annihilation predominates in Si, and the measured activation enthalpy of u has to be interpreted as Q = E g ~ / +2 w,. The Hirth-Lothe picture considered so far assumes that the kink motion is limited by W,. For weak W , Celli et al. (1963) argue that the motion of the kinks is significantly controlled by so-called “weak obstacles” (or dragging points), which are barriers distributed at random along the dislocation line with mean distance 1 and energy height E,. 1 is considered to exhibit a temperature de-


pendence like 1=lo exp(-e,lkT) describing either a thermal instability of the weak obstacles or their finite binding energy to the dislocations. The kinks have to overcome these weak obstacles by thermal fluctuations where waiting in front of the obstacles has significant effects on the mean kink velocity Dk and on the effective DK nucleation rate, J . It has to be taken into account for &z that the time for traveling the distance I , l/Dk, is given by the propagation time l / V k plus the waiting time vi1 exp (&/kT) (with vo=vD). For waiting times that are long compared with the propagation time this yields

& = 1 Yo exp ( - E d / k T )

(6-26)

The modification of the nucleation rate comes from the fact that stable DK’s cannot be created too close to a weak obstacle, since a kink waiting in front of an obstacle tends to run backwards by fluctuations in its diffusion way and to be annihilated with its partner. Within this model (generalized by Rybin and Orlow, 1970), Celli et ai. (1963) deduced the nucleation rate as follows:

J = ( V , / b )(1 +Ed/(bh 1 Z))

*

exp (-E,/(b h 1 t)- E & / k T )

(6-27)

Estimating the dislocation velocity from Eq. 6-25 with J and vk substituted by .i and ijk is the central point of the “weak-obstacle” theory of dislocation motion. The precise realization of the weak obstacles so far is not clear. Impurity atoms or atom clusters as well as lattice defects like vacancies or interstitials in the dislocation cores have been considered, as well as jogs and constrictions (see, e.g., Alexander, 1986). It seems that at present there are no convincing arguments particularly favoring one of these proposals, although there are arguments which make impurities or atom clusters and jogs or constrictions rather improbable candidates for weak obstacles in the case of freely moving

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straight dislocations with sufficiently large velocities (Alexander, 1986). In the earlier investigations one of the main arguments in favor of the weak obstacle theory was its ability to describe the complicated stress dependence of the apparent activation energy Q(z) := -d In v / d ( l/kT) as deduced experimentally, e.g., from v data obtained by the etch pit technique (cf. Sec. 6.5.2). According to the recent discussion of the experimental situation by Sumino (1 989), one has to be careful in using the dislocation velocity data from this technique. There are indications of a local pinning of dislocations in the surface regions sampled by this technique which are due to impurity clusters introduced from the surface when the crystal is kept at elevated temperatures to observe the dislocation motion. This pinning will be temperature dependent and particularly reduces the effective mobility of dislocations at low velocities, that is, at low stress t.In highly pure Si, as discussed by Sumino (1989), the dislocation velocity is linear in twith activation energy Q independent of z (as long as the shape of the moving dislocation remains “regular”). This has been confirmed in the stress range of 1 to 40 MPa and temperatures between 600” and 800°C with Q = 2.20 eV and 2.35 eV for 60” and screw dislocations in Si. Similar observations come from electron microscopy studies (Louchet, 1981; Hirsch et al., 1981). The electron microscopy investigations revealed a continuous motion of dislocations without any waiting events down to the resolution limit of the method of 5 nm which is interpreted as an indication that weak obstacles, if they exist, have a mean distance of less than 5 nm. The electron microscopy observations have been analyzed (see Jones, 1983) in terms of the Hirth-Lothe picture and the weak obstacle model, yielding in the former case a secondary Peierls potential

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6 Dislocations

W , of 1.2 and 1.35 eV. These estimates agree rather well with Jones's (1985) theoretical data from computer simulations mentioned above. This finding for Si, as well as the internal friction data of Jendrich and Haasen (1988) for Ge, might be considered to favor the Hirth-Lothe picture. As pointed out e.g. by Louchet and George (1983) for Si, the experimental values of v are larger than the predictions from this theory by some orders of magnitude, where the discrepancy was attributed to missing entropy terms. Following this idea Marklund (1 985) succeeded in showing by computer simulations that the vibrational part of the kink migration entropy has, indeed, the right magnitude to account for the missing factor, which gives additional support to the correctness of the Hirth-Lothe theory. An additional, necessary modification of the theory was introduced by Moller (1978) by considering that dislocations in Si and Ge are dissociated into partials with a stacking fault ribbon i n between, where the partials are tightly bound to the (1 10) Peierls valleys. Since screw dislocations dissociate into two 30" partials, 60" dislocations in one 30" and one 90" partial, DK nucleation and kink motion on these partials must be considered to be the fundamental processes. Following a suggestion by Labusch, Moller (1978) took into account that the DK formation on the corresponding partials should be correlated below a critical stress tc,but that a stresses above z, uncorrelated nucleation occurs. zc turns out to be yld, ( y : stacking fault energy, do: dissociation width of the two partials) where tc 10 MPa for 60" dislocations in Si, 16 MPa for screws (19 and 31 MPa in Ge). Moller (1978) compared his theory with etch pit measurements of the dislocation velocity in Ge (Schaumburg, 1972), Si (George et al., 1972), and GaAs (Choi et al., 1977). From this comparison he deduced estimates of the model parameters f:

as compiled and critically considered by Alexander ( 1986). In the light of Sumino's ( 1989) remarks concerning the etch pit technique, the meaning of the parameters is, however, somewhat unclear, and a discussion of Moller's findings should be postponed until the controversy about this technique is resolved.

6.6.2 Dislocation Motion in Doped Semiconductors

So far, we have considered theoretical approaches concerning the dislocation velocity in undoped material. There are a number of theories about the effects of dopants on dislocation motion. The most significant feature in the doping dependence of the dislocation motion is the observation that v increases with doping in n- and p-Si as well as in n-Ge whereas it decreases with doping in p-Ge. The present theories consider as possible sources of this doping effect (Patal effect) a doping dependence of the DK nucleation process or of the kink mobility. Particular examples of theories investigating the doping dependence of the DK formations are those by Patel and coworkers (Frisch and Patel, 1967; Patel et al., 1976), by Haasen (1975, 1979) and by Hirsch (1 979). The doping dependence of kink mobility is studied by Kulkarni and Williams (1976), by Jones (1 980, 1983), and by Jendrich and Haasen (1 988). Since the earlier literature has been reviewed in detail (e.g., by Alexander, 1986) basic ideas of only some of these theories will be presented here in order to reflect the present discussion in this field. Patel et al. (1 976) proposed that the kinks are associated mainly with special charged dislocation sites, and that any mechanism that increases the electron concentration will increase the density of charged dislocation sites and consequently raise the kink


concentration and dislocation velocity. In order to account for the observed effects, they assume that the dislocations in n- and p-type Ge as well as in n-Si introduce acceptor states whereas the dislocations are said to introduce donor-like states in p-Si. Patel and Testardi ( 1977a) succeeded in describing the relative change of the velocity of 60" dislocations in Ge by using a level position 0.13 eV below the conduction band. The best fit on screws in n-Si gave an acceptor level 0.6eV above the valence band maximum E,. Comparison with experiments on p-Si (at 450 "C) led to a donor level at the same position whereas velocity data for 550°C result in a donor level about 0.75 eV above E,. The approach was criticized by Schroter et al. (1977) since assuming centers of different type for dislocations in n- and p-Si yields severe inconsistencies if the doping is gradually lowered to the intrinsic range. Starting from n-doping, the theory predicts that the dislocations should be negatively charged in the intrinsic range. Starting from p-doping, it predicts a positive charge in this region. Patel and Testardi (1977b) admitted this difficulty and claimed that their theory at least holds for Ge and n-Si. A rather different approach was introduced by Haasen (1975,1979). He assumed that the dislocations have partially filled electronic perturbation bands in the gap, as predicted by the microscopic theory for unreconstructed configurations (cf. Sec. 6.4.2), and that they are able to carry a net charge. As source of the Patel effect he considers a change in the DK formation enthalpy caused by a change in the effective charge on the dislocations due to doping. In his model the dislocation line displacement created by the DK reduces the electrostatic energy of the arrangement. The energy gain turns out proportional to the square of the line charge and acts as a doping-dependent

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reduction of the DK formation enthalpy. The possible charge on a dislocation is limited because of the Coulomb self-energy, which limits the possible gain of formation enthalpy. Regarding this it seems that the resulting energy reduction might be too small in most cases to solely account for the observed doping dependence of the dislocation velocity. Hirsch (1979) in his theory treats dislocations which in their straight configuration have no deep energy levels in the band gap, as proposed by the microscopic theory for reconstructed core configurations (cf. Sec. 6.4.2). It is assumed that kinks have dangling bonds with deep donor and acceptor levels. Neutral and differently charged kinks are considered as independent thermodynamic species with individual equilibrium densities, where the density of charged kinks depends strongly on the relative position of the Fermi level compared to the kink acceptor or donor level E k A , Em. For the kink motion, Hirsch adopts the kink diffusion model with identical migration energy W , for charged and uncharged kinks. Consequently, the doping dependence of the dislocation velocity goes with the doping dependence of the total density of charged plus uncharged kinks. For n-type material, negatively charged kinks are of importance where the ratio between charged and uncharged kink density is determined by the position of E k A . The same seems to hold for p-Ge (which demands that EkA-E, not be too large, and that the temperature be high enough), whereas for p-Si positively charged kink donor states play the main role in giving their electrons to the chemical acceptors. For Ge Hirsch arrives at E k A - E, C 0.19 ev. AS reported by Jones (1983), Schroter has fitted earlier results in Si (George et al., 1972) to Hirsch's model yielding EkA- E, = 0.67 k0.04 eV and E,-E, = 0.28k0.17 eV.

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6 Dislocations

According to the discussion by Jones (1980), it seems likely that the low-energy kinks in their equilibrium configuration on reconstructed dislocations have no dangling bonds and consequently no deep-donor or acceptor levels but provide shallow levels, in contrast to Hirsch’s assumption. Regarding this observation, Jones (1980) deduced another model for the Pate1 effect. He assumes that in the process of migration, where in the saddle point configuration bonds may be stretched up to 30%, the shallow kink ground state levels deepen and hence, if charged, significantly lower the kink migration energy. In this picture. the kink density remains unaltered by doping, but the kink velocity changes. Within an atomistic model Heggie (1982) has shown that the kink levels i n Si fall to E,,-€, ~ 0 . 6 eV,E,,-E,=0.87 4 eV whenthekinks reach their saddle point which indicates that considerable changes to the migration energy W , are to be expected. Jones (1980) claims that the resulting expressions for the dislocation velocity in their formal structure agree with those of Hirsch (1979) although they have a different meaning and origin. Modified versions of the theory lateron are discussed by Jones ( 1983) now considering reconstruction defects (cf. Sec. 6.4.31, which he calls solitons or anriphase defecrs, on reconstructed dislocations as nucleation centers for DK’s. The reconstruction defects represent local dangling-bond centers with deep electron levels and their density, of course, will change with doping, inducing a doping dependence of the DK nucleation rate. This picture is based on Jones’s assumption ( 1 983) of a sufficiently low formation enthalpy for reconstruction defects. Another estimate (Teichler, 1989a) deduces a rather high formation enthalpy and hence a low equilibrium density for these defects. Accordingly, the question about the equilibrium density of the reconstruction

defects seems unsolved as far as theory is concerned. Experimentally (in Si), until now, no EPR signal of the isolated reconstruction defect has been detected despite intense efforts, which might be an indication that it is a rare configuration. In the same paper, Jones discusses that recombination of possibly highly mobile reconstruction defects with dangling-bond-carrying deep-level kinks, created during the DK nucleation process, may turn them into low energy, shallow-level kinks. The latter then exhibit a migration-energy reduction due to the deepening of the shallow levels in the saddle point configuration, as mentioned above. An important experimental fact, initiating a further model, was provided by the internal friction measurements on deformed Sb-doped Ge by Jendrich and Haasen (1988). They showed that the activation enthalpy of the lower-energy process, reflecting the motion of geometrical kinks, is reduced by 0.3 to 0.4 eV under doping with (2 to 4) x 10” cm-3 Sb. On the other hand, no doping dependence was observed in the activation enthalpy of the high-energy process attributed to kink pair formation. From a careful discussion of the available experimental and theoretical data, Jendrich and Haasen (1988) came to the conclusion that none of the models considered so far is able to provide a satisfying description of their experimental results. They propose that the doping dependence of the apparent W , may reflect a change from low-mobile reconstructed kinks to highly mobile unreconstructed ones, where the latter have deep electron levels due to their dangling bonds, and the density of the latter increases under n-doping because of an energy gain when filling their levels with electrons. Before closing this section, we should add one remark concerning the theories of dislocation velocity in undoped material.

6.7 Dislocation Generation and Plastic Deformation

The theories sketched above are constructed to describe rather idealized situations where single dislocations move through a rather perfect crystal. Sumino's (1 989) discussion about the pinning of near-surface dislocation segments at low stress by impurity clusters indicates that additional effects may be observed if the conditions are not as idealized as considered. A particular example of this is the effect of jogs introduced by climbing or by mutual cutting of dislocations from different glide systems. As proposed by Haasen (1 979), these jogs may act as "weak obstacles" for kink motion or as strong pinning centers for the dislocations. They thus may introduce additional new features in the theoretical picture of dislocation mobility in the case of macroscopic plastic deformation of the samples.

6.7 Dislocation Generation and Plastic Deformation 6.7.1 Dislocation Nucleation There are mainly three situations in which processors of semiconductors are faced with generation of dislocations: crystal growing, thermal processing, and growing epilayers on a substrate, leading to misfit dislocations. In the first part of this section, we will focus on the first two processes. In contrast to intrinsic point defects, dislocations are never in thermal equilibrium, because of the small entropy of formation compared to the large enthalpy. Consequently, it is possible in principle to grow crystals of any substance dislocation-free. Although in the case of ductile metals it is difficult to preserve such crystals without introducing dislocations by some surface damage, this is no problem with germanium and silicon, where dislocations are completely immobile at room temperature. Ac-

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tually, up to now, it was not yet possible to grow dislocation free crystals of A"'BV compounds without high doping. This fact is connected to the higher mobility of dislocations in those materials and to some technical complications arising with evaporation of one component. The existence of such difficulties and, moreover, generation of dislocations in originally dislocation-free material make it worthwhile to study conditions under which dislocations may be generated (Alexander, 1989). One idea involves nucleation of a dislocation loop by thermal fluctuations under the action of a mechanical stress. Obviously, introducing a dislocation means increasing the total energy of the crystal by the sum of strain and core energy of the loop. The driving force for nucleation of the loop is provided by the work done by the shear stress acting in the glide system of the loop; this work is proportional to the loop area. The resulting balance of energies goes through a maximum when the loop grows. Thus a critical radius R, and an activation energy E, are defined. Any reasonable estimate shows that E, by far exceeds thermal fluctuations at any temperature. This means that in a perfect crystal (including equilibrium vacancies) dislocations cannot be generated by any stress. We therefore must look for heterogeneous nucleation processes using some defects different from dislocations as nuclei. For semiconductors, the following are of importance: (1) Surface damage; (2) Agglomeration of native point defects; (3) Punching of dislocation loops at precipitates of a second phase. Before discussing these processes, it should be noted that Vanhellemont and Claeys (1988a), when dealing with yielding (i.e., nucleation and multiplication of the

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6 Dislocations

source dislocations), considered only processes (1) and (3) above (together with multiplication of grown-in dislocations) to be heterogeneous and process (2) to be homogeneous (yielding). 1) Surface damage comprises all processes where large local stress applied to a thin surface layer of the crystal causes relative displacement of two regions of the crystal. This happens for instance with scratching, grinding, hardness indentations, impinging hard particles, etc. Hill and Rowcliffe ( 1974) analyzed hardness indentations on silicon surfaces and came to the conclusion that locally the theoretical shear strength is overcome followed by an out-of-register recombination of the two faces of the cut. Temperatures above the brittle-ductile transition of the respective substance are required to expand the dislocation loops produced, which are of the order of 10 pm diameter. 2 ) Intrinsic point defects in excess of the thermal equilibrium density are produced either by cooling from a high temperature, especially during crystal growth, or by precipitation of some impurity species (e.g., oxygen in silicon) producing a huge amount of self-insterstitials (SI’s). Admittedly the equilibrium density of vacancies (V’s) and SI’s in semiconductors are considerably lower than i n metals, but any nonequilibrium concentration is hard to remove since annihilation of SI’S with V’s seems to be hindered by an energy barrier. Thus the common way to remove the excess point defects far from the crystal surface is agglomeration in spherical or - because of elastic strain energy - in platelike structures parallel to close-packed lattice planes. In the diamond structure, a double layer of V’s or SI’s embedded in the matrix is equivalent to an intrinsic or extrinsic stacking fault, respectively. We call this formation of an area of stacking fault (SF) step 1 of dislocation nu-

cleation by agglomeration. In compounds, V’s or SI’s of the two sublattices should coprecipitate to form an ordinary SF. But this ideal case in general will not be realized. If, for example, in GaAs excess arsenic atoms are precipitating interstitially, a full layer of interstitial GaAs is formed by emission of the related number of Ga vacancies. (The subsequent complicated steps leading to precipitates of hexagonal arsenic are not of interest here (B. F. Lee et al., 1988).) The SF mentioned above is bound by a sessile Frank partial loop. If, by climb of this loop, the disc reaches a certain critical size, a Shockley partial dislocation may be spontaneously nucleated in its center, removing the SF and transforming eventually the Frank partial into a loop of a perfect prismatic dislocation (unfaulting of the SF = step 2 of nucleation): a/3 [ 1 1 11 + a/6 [211]

+ a/2 [Ol 1 ] (in (1 11))

The Burgers vector of the resultant dislocation makes suitable segment of the loop glissile in either the (1 1I) or in the (1 11) plane (step 3). Here, a fundamental principle of all heterogeneous nucleation mechanisms comes to light: An expanding dislocation loop needs to oversome the backstress which is due to interaction between all loop segments. A rough estimate for a circular loop shows that the critical radius for expansion under a shear stress t i s R, = p b/2 t ( p b2/2 represents the line tension, p being the appropriate shear modulus). For typical values (Si: b = 3.85 X I O - ’ ~m, p = 63.4 MPa, t = 10 MPa) R, turns out as 1.25 pm. Considerable stress concentration is necessary to expand dislocation loops smaller than one micrometer. Thus the climb force (consisting of a chemical part due to the supersaturation of


point defects and possibly a mechanical part) must be sufficiently large to increase the loop to the critical size to that glide motion and multiplication of a loop segment becomes possible. In this case, the second period of dislocation generation begins, namely, multiplication, which will be treated in the next section. In a series of papers, Vanhellemont et al. analyzed the homogeneous nucleation of dislocations at the edge of a film (e.g., SiO,, SiN,, etc.) covering a silicon wafer (Vanhellemont et al., 1987; Vanhellemont and Claeys, 1988a, b). From the calculated stress field of the film edge a particular dislocation is determined, which for a given orientation of substrate and film edge will grow fastest by climb and subsequently by glide. For CZ-grown substrates the point defects nucleating the dislocations are SI’s produced by precipitating interstitial oxygen. The analysis is able to explain a great deal of various geometries and is confirmed by electron microscopical work. However, the origin of the first edge dislocation growing by climb is not clear. It is important to note that the stress field at the film edge promotes by mechanical climb force the precipitation of SI in this particular area and at particular dislocation types. FOll and Kolbesen (1975) showed that Aswirls in “dislocation-free” silicon consist of perfect extrinsic dislocation loops which are also nucleated by agglomeration of SI’s. Fall and Kolbesen found loops with SF’s only if carbon ( l O I 7 ~ m - and ~ ) oxygen ( 10l6~ m - were ~ ) present in high concentrations. Apparently, these impurities stabilized the stacking fault against unfaulting. The observed loops are rather large (0.5 to 1 pm). From their density (lo6- IO7 ~ m - ~ one may deduce a dislocation density of the investigated “dislocation-free” crystals of about IO2- lo3 cm-*. Only part of those dislocations becomes mobile by the release

367

of a segment of the prismatic loop onto a glide plane. One may calculate from the data given in the paper that between lOI3 and 5 x l o t 4 cm-3 of SI’s are removed from the crystal by formation of A-swirls. It should be stressed that the tendency of point defects to agglomerate in semiconductors is strongly correlated to their charge state and, therefore, to the actual position of the Fermi level. Convincing proof for that was given by Lagowski et a]. (1984) in a paper treating the strong influence of electrical doping on the density No of grown-in dislocations in GaAs crystals grown by the horizontal Bridgman technique. While No for undoped GaAs is of the order of lo3 ~ m - it ~ ,decreases in n-type material (due to doping with Si or Se) to effectively zero and increases in p-type (Zn) crystals to 5 x lo4 ~ m - Actually, ~. only the net doping ( N D - N A ) is of influence. Since the arsenic vacancy (VAS)is positively charged wherever the Fermi level is, it must be the V,, whose charge state determines the agglomeration of double layers of vacancies as the first step of dislocation generation under the low-stress conditions of the particular growth technique. The gallium vacancy is an acceptor with several states in the lower half of the energy gap. The authors claim that in a case where gallium vacancies dominate, arsenic antisite defects (AsGa) are generated and not arsenic interstitials, to complete a double layer of vacancies. The reaction proceeds as follows: V,;

+ AsAs + AsGa + VLS+ 3 e-

The transition of an arsenic atom into a gallium vacancy (thereby providing the arsenic vacancy) is more promoted with de) creasing numbers of free electrons. This means that the first step of dislocation nucleation by agglomeration of supersaturated vacancies in GaAs proceeds in proportion to (n)-3, which is in good agreement

368

6 Dislocations

with experiment. Generation of AsGaantisite defects together with dislocations can be understood in this model as well. 3) Particles of a second phase (e.g., oxides) may produce dislocation loops in the surrounding matrix either by a volume misfit or by a difference of thermal expansion between particle and matrix. As was shown by Ashby and Johnson ( 1969), glide dislocation loops are generated around spherical particles in the glide plane of maximum shear stress. In many cases, the screw segments of those loops disappear afterwards by repeated cross slip, leaving prismatic dislocation loops. The transformation from a glide to a prismatic loop does not always take place: in dislocation-free FZ Si, a high shear stress (200 MPa) applied for some hours at 420°C (after pretreatment at 700°C) produces large glide loops in the plane of maximum shear stress (Kruchten v., 1984; Alexander et al., 1983). Calculating the critical radius for loop expansion under 200 MPa shear stress, one finds that the radius of the original loop, and therefore, of the particle, must be at least 70 nm. Referring to crystal growth under liquid confinement (liquid encapsulation Czochralski technique), precipitations of one component of the compound or inclusions of the encapsulant glass may nucleate dislocations. For expansion of those nuclei of loops, the stress-temperature history of the crystal is decisive. This expansion and multiplication belongs to the second period of dislocation generation: growth and multliplication. Concluding this section, we should mention another type of dislocation nucleation which occurs frequently in metallic alloys: constitutional supercooling. Depending on the distribution coefficients k , in the system under consideration, on the temperature gradient at the solidification interface, and on the growth rate, local supercooling may destabilize the planar solidification

interface, and the freezing crystal will then be divided into cells separated by cell walls enriched in one of the components of the alloy. Those chemical inhomogeneities are connected with differences of the lattice constant and may eventually lead to small angle boundaries of misfit dislocations. It has been discussed whether the well-known cell structure of LEC-grown GaAs might be due to constitutional supercooling. But the observation that dislocation cells can be generated by after-growth anneal points to polygonization of otherwise produced dislocations. 6.7.2 Dislocation Multiplication

(Plastic Deformation) The dislocation content No of as-grown crystals is often explained by plastic deformation under the action of thermal stress during the cooling period. This idea has its origin from the observation of dislocation etch pits on the crystal surface and on cross sections being arranged along slip lines, i.e., along the traces of slip planes. In Sec. 6.7.1 it was shown that this does not provide a complete explanation. First some “source” dislocations have to be nucleated from defects of a different kind, whereby stress is helpful. Not before a critical density of such mobile dislocations (of the order of IO3lo4 cm-2) is nucleated can motion and multiplication of dislocation -Le. plastic deformation - take over the increase of the dislocation density (by several orders of magnitude). For improving growth methods, this distinction seems to be important, because it demonstrates that it may be more promising to control intrinsic point defects and precipitation of oxides than to remove fully thermal stress. Crystal cooling and its influence on plastic deformation is often tackled in the framework of the model of an elastic-plastic solid. Here it is assumed that


any volume element of the crystal may accommodate elastically a certain part aelof the thermal stress, the excess stress being removed by plastic strain with a linear relationship between and (a-a,,). Calculating from the tensor of thermal stress the shear stress t in the most-stressed glide system, telis called “yield stress” or “critical resolved shear stress CRSS”. The growth conditions are then adjusted so as not to reach the CRSS in any part of the growing crystal. This conception may be accepted for f.c.c. metals with “instantaneous” response of dislocations to the stress distribution. However, because of the thermally activated dislocation motion in semiconductors (cf. Sec. 6.5), the thermal history of the considered volume element plays an important role. Here the relation between shear stress and plastic strain is far from being unique. Moreover, the same & may be ac*! comodated by a few fast-moving dislocations or by many slow ones. Thus the dislocation density that eventually appears depends on stress and temperature during the whole cooling history. The conception of CRSS therefore suffers mainly from its neglecting the dimension of time, not so much from the (not generally correct) assumption of a starting stress for dislocation activity. Semiquantitative analysis of density and distribution of grown-in dislocations in asgrown crystals of InP (LEC grown) on the basis of the dynamic properties of dislocations in that particular semiconductor has been carried out by Volkl (1988) and Volkl et al. (1 987). Data on the properties of dislocations are obtained by standard deformation tests: single crystals are compressed uniaxially along an axis far from any highly symmetric direction. In this manner, mainly one glide system is activated: about 80% of the dislocations belong to one Burgers vector and glide plane (single slip). Beneficial for those tests are crystals with

369

about lo4 cm-2 grown-in dislocations so that nucleation of dislocations does not interfere. The compression test may be carried out with constant strain rate & (the dynamical test resulting in a stress-strain curve) or with constant shear stress t (creep test). An analysis of such deformation tests was carried out in the 1960s and is reviewed elsewhere (Alexander and Haasen, 1968; Alexander, 1986). Here we give just the essentials: Applying Eq. 6-16 for the dislocation velocity v (t,T ) to plastic deformation, Le., simultaneous activity of very many dislocations, one has to replace the shear stress tapplied to the crystal by an effective stress teff= t-A

fl

(6-28)

(N: actual dislocation density. For single slip, the term A fl stems mainly from parallel dislocations of the primary slip system and can be calculated from the theory of elasticity). Equation (6-28) describes screening of the stress z by the stress field of the other dislocations. To calculate the development of the dislocation density N during deformation one has to know the law of dislocation multiplication. Experiments with Ge and Si revealed as a reliable approximation:

dN = N K

t&

v dt

(6-29)

( n being I or 0). From Eq. (6-29), it becomes clear that multiplication proceeds by motion and not from fixed sources like the FrankRead source. For compounds, this may be different because the segments of dislocation loops have extremely different mobility so that at the beginning of deformation only suitably oriented surface sources are active (Kesteloot, 198 1). Thenature of those sources is not well understood; it is possible that they are due to surface damage. The stress exponent n in Eq. (6-29) depends on the dislocation density: for weakly deformed crystals, n is zero (no explicity

370

6 Dislocations

dependence on stress); in a heavily deformed state, n becomes l (the extension of dipoles now dominates dislocation multiplication) (Alexander, 1986). Combining Eqs. (6-16), (6-28), and (6-29) with the Orowan relation Epi

=N b v

(6-30)

which treats plastic strain rate as flux of mobile dislocations, one may calculate the dislocation density N and either stress z (for the dynamical test) or plastic strain (creep test) as a function of time for a given temperature. The Eqs. (6-16) and (6-28)-(6-30) offer an easy approach to the yield point phenomenon of dislocation-lean crystals (Johnston and Gilman, 1959). As long as N is small (< 10' crn-*), the experimentally prescribed strain rate & has to be provided mainly by elastic strain & = ZIG

+N b u

(G: effective shear modulus of the specimen and the machine). Thus the stress increases with time quasielastically. Simultaneously, the dislocation density increases up to the upper yield point of the stress-strain curve where for the first time ( N b u ) equals &. Continuing deformation further increases N , and stress can decrease. This causes the elastic term to become negative: the stress-strain curve adopts a negative slope. Eventually, at the lower yield point ( N b u) again equals 6 but now with a dislocation density about three orders of magnitude bigger than at the upper yield point. From the lower yield point on, strain hardening dominates the stress, and the stress-strain curve follows the same three-stage scheme as with many metals. The analysis of the first steps of plastic deformation as it was described above works so easily because most of the dislocations which become immobile during deforma-

tion are present as multipoles and therefore do not significantly contribute to the screening of stress, either. To calculate the (local) dislocation density that results from plastic deformation in a crystal cooling from the freezing temperature, we proceed as follows: first, a reasonable density of nucleated source dislocations of critical length is chosen, and then each volume element of the crystal is followed as it moves through the particular temperature field of the particular apparatus for crystal growing. Of course, the growth rate introduces the dimension of time. Interestingly enough, not only slow growth may result in good crystals (because here the thermal stess may be better controlled) but also rather large growth rates again may be suitable: This time is too short for extensive dislocation multiplication before the considered volume element comes into cooler regions where dislocations move slowly. This variant is used for growing dislocation-free silicon, but for A"'BV compounds, with dislocations being more mobile, the necessary growth rate is too high to be realized at the present time. 6.7.3 Generation of Misfit Dislocations

One of the most frequent steps in device fabrication involves growing a layer of semiconducting material on top of a crystalline substrate with a different lattice parameter. It is clear from first principles that a critical thickness t , of the layer will be present, beyond which the introduction of dislocations with an edge component into the interface region reduces the total energy of the system. t, will depend on the lattice mismatch 6. However, experimental evidence shows that dislocation-free layers thicker than t, often grow on a dislocation-free substrate. One must therefore conclude that the nucleation of misfit dislocations must overcome some barrier as well.

6.9 References

In fact, modern theories (Dodson and Tsao, 1987; Hull et al., 1989) do not consider an equilibrium situation but rather relaxation from a metastable dislocation-free state by nucleation and propagation of misfit dislocations. Applying the microdynamical theory as out-lined in 6.7.2 or similar models, these theories give calculations of the density of misfit dislocations to be expected for a given thermal history of the multilayer system. “A crucial factor determining strain relaxation is time at temperature as well as layer thickness” (Hull et al., 1989). Hull et al. (1989) describe for the system Ge,Si,,/Si a method of measuring the material parameters needed for calculation, with respect to plasticity in the electron microscope. As in all computations along the lines of the microdynamical theory, the origin and the density of the first source dislocations must be known. One possible mechanism for the formation of such sources has been described by Hagen and Strunk (1978). Actually, there is no generally accepted model for the origin of the first misfit dislocations for those cases where the substrate does not provide threading source dislocations. (For a recent discussion stressing the nucleation of partial dislocations compare Hirsch, 1991.) For misfits exceeding 2% homogeneous nucleations of dislocation half loops at surface steps has been proposed (Eaglesham et al., 1989). For lower misfit, these authors quite recently found a new type of dislocation source. It consists of a particular type of stacking fault whose bounding partial can dissociate into a Stockley partial and a perfect dislocation. In fact, perfect dislocations on two different glide planes can be produced and the process is self-reproducing. These properties are deduced from electron microscopy of Ge,Si,, layers on Si (100). It must be left for future research whether this type of source is frequent.

371

6.7.4 Gettering with the Help of Dislocations Frequently during high-temperature processing of silicon devices, metal atoms precipitate, mostly as silicides; these precipitates have to be removed afterwards. As was shown by Ourmazd (1986), one may achieve that in principle by a two-step process: First, the precipitates are dissolved by apulse of self-interstitials, and then the metal atoms are fixed (gettered) in a strained area far from the active region of the device. Strain-causing gettering mechanisms may be produced by surface damage (either on the front or on the back side), by phosphorous diffusion, by film deposition or by a defective layer in the bulk some 20 ym below the active zone. This latter technique is known as “intrinsic gettering”. In fact, dislocation generation is induced by oxide precipitation. Before those oxides are formed, the oxygen has to be outdiffused from the active layer.

6.8 Acknowledgement The authors wish to thank W. Schroter for critically reading the manuscript.

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7 Grain Boundaries in Semiconductors Jany Thibault. Jean-Luc Rouviere and Alain Bourret CEA.Grenoble. DCpartement de Recherche Fondamentale sur la Matikre CondensCe. 17 rue des Martyrs. Grenoble. France

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 379 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 7.2 Grain Boundary Structure: Concepts and Tools . . . . . . . . . . . . . 383 7.2.1 Grain Boundary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 384 385 7.2.2 Geometrical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Dislocation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 388 7.2.3.1 Primary Dislocation Network . . . . . . . . . . . . . . . . . . . . . . . . 389 7.2.3.2 Secondary Dislocation Network . . . . . . . . . . . . . . . . . . . . . . 7.2.3.3 Stress Field Associated with Grain Boundaries . . . . . . . . . . . . . . . 390 392 7.2.4 Structural Unit Descriptions . . . . . . . . . . . . . . . . . . . . . . . . 392 7.2.4.1 Stick and Ball Structural Units . . . . . . . . . . . . . . . . . . . . . . . 393 7.2.4.2 Energetic Structural Units . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4.3 Algebraic Structural Units . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.2.4.4 Structural Units and Dislocations/Disclinations . . . . . . . . . . . . . . 395 7.2.4.5 The Limits of the Structural Unit Descriptions . . . . . . . . . . . . . . . 395 7.2.5 Computer Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . 395 7.2.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 396 7.2.5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 7.2.5.3 Interaction Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 7.2.6 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Grain Boundary Structure: Experience and Simulation Results . . . . 401 7.3.1 Silicon and Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 401 7.3.1.1 Tilt Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1.2 Twist Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 414 7.3.2 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Sic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 415 7.3.4 GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 415 7.3.6 A1N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.7 NiO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 7.3.8 Comments on Grain Boundary Structures . . . . . . . . . . . . . . . . . 416 7.4 Electrical Properties of Grain Boundaries . . . . . . . . . . . . . . . . 417 7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 7.4.2 Electrical Effects Induced by Grain Boundaries . . . . . . . . . . . . . . 417 7.4.2.1 Electronic States Associated with a Grain Boundary . . . . . . . . . . . . 417

378 7.4.2.2 7.4.2.3 7.4.3 7.4.3.1 7.4.3.2 7.4.4 7.4.4.1 7.4.4.2 7.4.4.3 7.4.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.3 7.5.4 7.5.4.1 7.5.4.2 7.5.4.3 7.5.5 7.6 7.6.1 7.6.2 7.6.2.1 7.6.2.2 7.6.2.3 7.6.2.4 7.6.3 7.6.3.1 7.6.3.2 7.6.4 7.7 7.8

7 Grain Boundaries in Semiconductors

Potential Barrier and Transport Properties . . . . . . . . . . . . . . . . . 421 Dynamic Properties and Recombination Properties . . . . . . . . . . . . . 424 Experimental Methods for Measuring the Grain Boundary Electrical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Methods Based on Transport . . . . . . . . . . . . . . . . . . . . . . . . 424 Transient Met hods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Correlation Between Electrical Activity and Structure . . . . . . . . . . . 426 426 Transport Experiments in Bicrystals . . . . . . . . . . . . . . . . . . . . Transport Properties Measured on Bicrystals . . . . . . . . . . . . . . . . 427 Emission and Capture Properties of Silicon and Germanium Grain Boundaries . . . . . . . . . . . . . . . . . . . . . 428 Polycrystalline Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Intrinsic or Extrinsic Origin of Electrical Activity of Grain Boundaries . . 429 Impurity Segregation and Precipitation Induced by Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Dopant Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Oxygen and Sulfur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Transition Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 434 Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Properties of Grain Boundaries in Semiconductors . . . . 435 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Interaction Between Dislocations and Grain Boundaries . . . . . . . . . . 436 Dislocation Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Dislocation Transmission Across Grain Boundaries . . . . . . . . . . . . 439 Grain Boundaries as a Dislocation Source . . . . . . . . . . . . . . . . . 440 Grain Boundary Dislocation Movement . . . . . . . . . . . . . . . . . . 440 Physical Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Grain Boundary Migration . . . . . . . . . . . . . . . . . . . . . . . . . 441 Recovery of the Grain Boundary Structure and Cavitation . . . . . . . . . 442 Deformation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445


List of Symbols and Abbreviations crystallographic lattice parameters Richardson constant arbitrary function; name of SU Burgers vector modulus of the Burger vector b pseudo Burgers vector of a SU secondary dislocation Burgers vector characteristic frequency for carrier capture separation distance between dislocations period of superposed set of secondary dislocations electron charge energy core energy of a dislocation energy of a dislocation Fermi level characteristic frequency for emission frequency energy barrier height Fermi distribution function function punctual operation step height; location of the GB along the GB normal or Planck constant Miller indices identity function or matrix current vector of structural unit A, B Boltzmann constant coefficient or segregation coefficient prime integers box sizes in simulation effective mass mobility of the GB normal of the GB integers integer; concentration of majority carrier; doping level density of states arbitrary vectors or period of the GB interface charges at the GB R , R,, R, ,R,, rotation or rotation matrix ro, ro2 vectors of the 0-lattice and 02-latttice r0 core radius of a dislocation ri radius of cylinder over which the energy of the dislocation is calculated S total Burgers vector

379

380


vectors measuring the length of a SU respectively in crystals I and I1 displacement of the CSL in grains I and I1 time a vector of the DSC lattice temperature melting temperature a vector of lattice I zone axis indices crystal symmetry operator voltages GB velocity thermal velocity coordinate vector of a lattice point of crystal I expressed in lattice I base coordinate vector of a lattice point of crystal I1 expressed in lattice I axes. y z is the GB plane mean atomic number inverse of a power of two (generally 1 or W) GB energy per unit of interface area and energy coefficient dielectric constant integers equal to -1, 0 or 1 rotation angles natural rotation in the SU-dislocation analysis shear modulus Poisson's ratio rotation axis modulus of the rotation axis vector conductivity shear component coincidence index: surface energy rigid body translation relaxation time activation energy relaxation parameters b.c.c. BV CBED CSL CVD dcc dCP DIGM DLTS DSC

body-centered cubic Burgers vector convergent beam electron diffraction coincidence site lattice chemical vapor deposition dichromatic complex dichromatic pattern diffusion-induced grain boundary migration deep level transient spectroscopy displacement shift complete or displacement symmetry consensing


EBIC EELS ELNES ESR f.c.c. FIM FS/RH GB GBD HREM IBIC IDB LBIC LCAO RBT STEBIC STEM STM

su

TDB TEM TFT UHV

electron beam induced current electron energy loss spectroscopy energy loss near edge spectroscopy electron spin resonance face-centered cubic field ion microscopy final stadright hand grain boundary grain boundary dislocation high resolution electron microscopy ion beam induced current inversion domain boundary light beam induced current linear combination of atomic orbitals rigid body translation scanning transmission electron beam induced current scanning transmission electron microscope scanning tunneling microscopy structural unit translation domain boundary transmission electron microscopy thin-film transistor ultrahigh vacuum

381

382


7.1 Introduction Grain boundaries (GBs), which are interfaces between two crystals that differ only by their orientation, are almost always present i n natural crystalline materials. They can greatly influence the properties (mechanical, electrical conductivity, optical emission, etc.) of the material in which they lie. Consequently, most of the semiconductor industries have initially tried to avoid them, and worked either with single crystals elaborated by techniques such as Czochralski growth, fusion zone, etc., or with monocrystalline thin films of low defect density. However, such crystals or layers are expensive and it turned out that polycrystalline materials could also play an important role in the semiconductor industry, either due to their low cost (as in solar cells) or due to their specific properties (for instance, the preferential diffusion and doping in polycrystalline regions of transistors), or because of the impossibility to do something else (in diamond thin films). It is forecast that in future years the production of polycrystalline silicon could be as high as the production of monocrystalline silicon. Thus silicon polycrystals were an important material to study but it turned out that they contain a lot of impurities. Consequently, the academic studies on semiconductor grain boundaries really started in the 1980s with the elaboration of pure silicon or germanium bicrystals (essential tilt GBs) and the use of electron microscopy. Beside important topics such impurity segregation and deformation, the big academic issues were: Is the interface amorphous or crystalline? Is the grain boundary itself electrically active? As we will see, clear answers have been given to these questions for high coincidence tilt GBs in silicon. However. new studies on twist GBs or nanocrystals have argued whether these conclusions could be extended to “general GBs”.

In this chapter, we will focus on academic studies of GBs in semiconductors. The chapter has five main parts. Sections 7.2 and 7.3 cover the GB structure, Secs. 7.4, 7.5, and 7.6 deal with the electrical activity, the impurities, and the mechanical deformation of semiconductor GBs, respectively. Section 7.2 is devoted to the different theories or tools that can be used to analyze or determine the GB structure. Thus after some definitions, we introduce some geometrical tools, which are based essentially on the symmetry relationship between two lattices of the same material. These tools are powerful means to describe periodical interfaces, symmetry breaks, and defects associated within GBs, although they provide no information about stress fields or atomic positions. Then, the description of a GB in terms of dislocations is presented. This was the first approach to ascertain the stress fields around a GB, and the application of elastic theory was of great help in explaining a lot of GB behavior (migration, interaction of GBs with other defects). However, this elastic approach does not work for high-angle GBs. We follow this with a description of GBs in terms of structural units, which are based on atomistic structures, but can be related to dislocations. This approach allows the construction of many atomic models of a given GB and can predict, through the principle of continuity, the most probable structure. It can be applied to any low or high angle GBs. Then there is a quick presentation of the numerical and experimental tools that have been used in the study of GB structures. Atomistic computer simulations have become one of the most powerful tools to analyze both the GB atomistic and electronic structures. High resolution electron microscopy (HREM) has been the major experimental tool to study the atomic structure of GBs, but new possibilities have emerged with the advent

7.2 Grain Boundary Structure: Concepts and Tools

of scanning tunneling microscopy (STM), atom probe field ion microscopy (FIM), and EELS (electron energy-loss spectrosCOPY).

Section 7.3 onwards summarizes the experimental and simulation results obtained on different GB atomic structures in different semiconductors. Each semiconductor is treated in turn. Silicon has been studied the most. Recently, new studies on silicon have emerged with the possibility to form nanocrystals or develop wafer bonding (which stimulates twist GB studies). But GBs in other semiconductors such as ionic semiconductors (NiO) or wide band semiconductors (diamond, silicon carbide and to a lesser extent GaN and AlN) have appeared. The last few years have seen an increase of GB simulations. Whereas for years in metals most GB simulations were calculated with only a few experimental HREM results being available for comparison, the reverse situation was true for tilt GBs in elemental semiconductors. However, simulations have again taken the lead as far as twist GBs and nanocrystals are concerned. This part of the chapter ends with a discussion on the validity and generalization of all these results. Section 7.4 onwards is devoted to the electrical properties of GBs in semiconductors. These GBs may drastically affect the electronic response of semiconductor materials, and special attention is put on additional defects or impurities segregated at the GB, since these are the most likely origin of GB electrical activity. Section 7.5 onwards is devoted to new results available on the segregation occurring at GBs, and Sec. 7.6 onwards describes the deformation of crystals containing GBs and the subsequent GB modifications. The strong interaction between point and linear defects is shown to have a major influence on the behavior of GBs. Emphasis is placed

383

on some of the consequences of these events, such as GB migration, enhanced diffusion along migrated GBs, and GB recovery and cavitation. It should be pointed out that at the end of this chapter several general references are provided to which the reader is invited to refer for completeness. They provide an overview of GBs in materials in general and in semiconductors in particular.

7.2 Grain Boundary Structure: Concepts and Tools Grain boundaries (GBs) can be studied for their specific properties in diffusion, impurity absorption, deformation, and electronic properties. But a thorough analysis starts with a full description of the defect itself its definition and its structure. So in this section, after some basic definitions, we present different concepts or tools, both theoretical and experimental ones which can be used to analyze the GB’s structure. The theoretical concepts are separated into three types: the geometrical concepts, the dislocation model, and the structural unit descriptions. Historically, these different approaches have been developed to describe GBs, and predict or understand some of their properties. However, we will see that there is a strong correlation between these approaches (for instance, the structural unit model with the dislocation approach, and the simulations with the 0-lattice and the dislocation content). All of them bring specific information on the GB structure, and they are in fact greatly correlated and complementary. This part ends with sections recalling the simulation techniques and the experimental tools that have been used to study the GB’s structure.

384


7.2.1 Grain Boundary Definitions A grain boundary is a two-dimensional

defect that separates two disoriented grains of the same crystalline material (see Fig. 7-1). These two grains can be related by an affine transformation (G, t),where G is a punctual operation and t is a rigid body translation (RBT). Thus each lattice point of crystal 11, whose coordinate vector i n lattice I base is X,,, is the image of a lattice point of crystal I (X,) X , , = G x,+t

( 7 -1)

This affine transformation is not unique. In symmorphic material (with neither glide nor screw operation), G can be chosen to be a proper rotation, which we will designate R in this chapter. In nonsymmorphic material, G can be broken down into the product of a proper rotation and an inversion (In?.Tables of Cnstalfography, 1983). When the punctual operation can be reduced to the identity operation, these defects have been called stacking faults or antiphase

Figure 7-1. Scheme illustrating the creation of a grain boundary (GB). The GB plane is gray. The two dotted parallelograms are references for, respectively, the front face of crystal I1 and for the interface; they make it possible to visualize the nine parameters that define the GB. Crystal I1 is deduced from crystal 1 by a rotation R (determined by its rotation axis e and its angle 8 : 3 parameters) followed by a translation 7 ( 3 parameters). The planar interface is defined by its normal n ( 2 parameters) and its location h along its normal ( 1 parameter).

domain boundaries or stacking mismatch boundaries depending on their atomic structure or on their origin. They should be better designated under the generic term translation domain boundaries (TDBs) (Rouvikre et al., 1997), as the only operation linking the two crystals is a translation. Generally, these TDBs are not considered as “real” GBs, perhaps because some of them can be obtained in a unique grain by the dissociation of a dislocation. Thus we will not consider them in this chapter. When the simplest punctual operation relating the two crystals is an inversion operation, the defect is designated as an inversion domain boundary (IDB). In that case, two grains exist and IDB must be considered as a special type of GB . Depending on the geometry of the interface, a GB can be flat, faceted, or curved, although it must be noted that these latter characteristics generally depend on the scale of observation. In this chapter, we will consider only flat GBs. Thus, in addition to the affine transformation, a flat grain boundary is defined if the normal n of the boundary (n points from crystal 1 to crystal 11: throughout the paper, crystal I is the reference lattice) and the location h along n of its interface are known (Fig. 7.1). Thus there are nine parameters to describe a GB. Five parameters may be varied independent of the others and define the macroscopic degrees of freedom ( R and n). The four other ones (t,h ) are called the microscopic parameters, because they are determined by relaxation processes and are thus a function of the macroscopic ones if we assume that there is only one energy minima for a given set of macroscopic parameters. When n is perpendicular to the rotation axis, the GB is called a tilt GB, whereas if n is parallel to the rotation axis, it is called a twist GB. Mixed GBs have both a tilt and a twist component.

385


A symmetrical GB has its interface in a symmetrical position with respect to both grains, that is to say, if the interface normal n is equal to the vector [ k , , k2, k,], in grain I, it will be represented by a vector ( k , , k2, k3)IIin grain 11.

volumes of the unit cell of the CSL and of the crystal lattice. In other words, considering one of the two original lattices, l/Zrepresents the density of lattice sites that are coincident to both lattices. In the following, a symmetrical GB will be denoted in the form

7.2.2 Geometrical Concepts

(hk l)(u v w ) Z = N

The several important geometric concepts presented in this part can be put into three categories. Some, like the coincidence site lattice or the coincidence index, determine the type of disorientation between the two grains and can be regarded as an extra set of definitions for the GB. Some, like the 0-lattice, will be useful in the description of the GB in terms of dislocations. Some define the symmetry of a bicrystal and are useful in determining all the possible variants of a GB and analyzing the appearance of GB defects arising from a break in the different levels of symmetry. Of course, all these geometrical concepts say nothing about the atomic local relaxations at the interface or about the energy of the GBs. The coincidence site lattice (CSL) is the intersection of the lattices of the two adjoining crystals. Friedel (1926) first introduced this concept in order to describe twins. Brandon et al. (1964) confirmed the validity of the CSL idea with the use of field ion microscopy. This approach was first applied to cubic lattices and then extended to nonexact coincidence positions and to noncubic structures in which the dimension of the common lattice might be reduced, leading to a 1 D, 2 D, or 3 D CSL. If lattice vectors are considered, there is no problem of origin. However, if lattice points are considered, the origins of the two lattices should be common. For some rotations R a threedimensional CSL exists and it can be described by an integer called the coincidence index Z, which is defined as the ratio of the

where N is the Zvalue (integer), = (u LJ w) represents the rotation axis of the proper rotation R and ( h k I ) refers to the GB plane, both expressed in crystal I. In the case of an asymmetrical GB, the coordinates of the GB plane in crystal I and crystal I1 will be given. In the cubic system, 2 is always an odd integer and Ranganathan (1966) showed that the Zvalue, the rotation axis = ( u LJ w) and the rotation angle 8 are linked by the following formulas

(7-2)

e

e

where k l , k2 are prime integers, @ =I =

Jw,

and a is the inverse of a power of two (generally 1 or 0.5) calculated such that 2 is odd. Equations (7-3) entirely determine the rotation matrix R . In a base where the third basis vector is parallel to 0 , the matrix R is given by

R=

1 . k f + k ; p2

(7-4)

k f - k ; p2 - 2 k l k 2 p

0

Tables giving the coincidence indexes, the CSLs, and the associated rotations of GBs in cubic systems have been established by Grimmer et al. (1974), and by Mykura (1980). Bleris et al. (1982) studied hexagonal systems and generated tables where the

386


cla ratio was taken into account, and pro-

posed a method to determine the exact relationship experimentally (Bleris et al., 198 1). A complete treatment of CSLs in hexagonal materials has been given by Grimmer (1989). and this was extended to rhombohedral and tetragonal lattices by Grimmer and Bonnet (1990). For some families of symmetrical tilt GBs in the diamond structure, GBs have been described by a set of two signed prime integers ( k , , k 2 ) , which verifies Eqs. (7-3) and (7-4) (Moller, 1982; Koyhama, 1987; Rouvikreand Bourret, 1990a). Here we adopt what we think is the simplest convention. To every prime integer ( k , ,k Z ) ,we associate a unique symmetrical tilt GB (where is the rotation axis) whose normal in lattice I is [ w k , ,M s k Z , - ~ k , - i ~ k For ~ ] ~the. [0, 0, 11 and [0, 1, 11 tilt GBs, this formula reduces to [ k , , k , , 01, and [ k , , k Z ,- k , ] , , respectively. This formulation synthesizes all the behavior of a series of GBs in a few formulas and, as we will see, helps in determining all the different types of GBs in a given series. Moreover, it shows that, for a given rotation axis, the macroscopic degrees of freedom of a high-coincidence symmetrical tilt GB are reduced from five to two parameters k , and

e

k2.

For the asymmetrical GBs, Paidar (1992) developed a systematic geometrical classification depending on the disorientation angle and the inclination of the GB plane. Bollmann (1970) introduced several important lattice definitions which are very useful when the GB is analyzed in terms of dislocations. The 0-lattice, which is an extension of the CSL, is the set of points - and not only lattice points - that have an equivalent position in the two crystals. The O-elements are given as a solution of the basic equation (Z-R-l) ro=fl

(7-5)

where t , describes all the possible translation vectors of lattice I and Z is the identity. Here, this equation defines the 0-lattice. It can also be used to define the dislocation content of a GB. Bollmann developed this approach for a general interface, where 0elements can be points, lines, or planes. In the simple case of a GB described by rotation (we neglect here the case of noncentrosymmetric crystals), the 0-elements exist for any disorientation angle 8 , even when no CSL exists and they are lines parallel to the rotation axis. In the case of a cubic crystal, the 2D lattice of these 0-lines in the plane normal to e can be simply determined by applying Eqs. (7-4) and (7-5) (Bollmann, 1970; Christian, 1981).The 0-elements can be considered as regions of good fit between the two adjoining lattices. These 0-element regions are separated by cell walls whose intersections with the GB plane represent the dislocation lines of the primary dislocation network [this is the b-net introduced by Bollmann ( 1970)]. For high-angle boundaries (characterized by a rotation R ) , with a small deviation from an exact coincidence position (characterized by a rotation Rr), Bollmann (1 970) also introduced the DSClattice and the 02-lattice. The DSC-lattice is obtained by considering all the linear combinations of the vectors belonging to the two lattices I and 11. The original meaning of the DSC acronym, which is Displacement Shift Complete, is not straightforward. As pointed out by Pond and Bollmann (1979), it should be replaced by Displacement Symmetry Conserving, as any translation of lattice I by any vector of the DSC-lattice conserves the original CSL-lattice. Originally, the 02-lattice acronym, which means second order 0-lattice, was defined as the 0lattice between the two 0-lattices (and not between the two crystal lattices) obtained by using Eq. ( 7 - 5 )once with R (0-lattice of the studied GB) and once with R, (0-lattice of


the reference high coincidence GB near the studied GB). However, the 02-lattice can also be defined as the 0-lattice between the two DSC-lattices associated respectively with the rotations R , and R , which we respectively designate DSC, and DSC. To obtain the equation verified by the 02-elements, it is then only necessary to determine the rotation matrix 6 R linking the DSC, to the DSC ( 6 R = R R;') and apply Eq. (7-5). The 02-elements are thus the solutions of the equation

(Z-6R-') yo*= (Z-R, R-') r o 2 = t D S C (7-6) where tDsc describes all the vectors of the DSC lattice. We will see in Secs. 7.2.3.2 and 7.2.4.3 that Eq. (7-6) is useful to analyze the secondary dislocations. Grimmer and Bonnet (1 990) applied the DSC approach in their treatment of rhombohedral and tetragonal materials. Pond and Bollmann (1979) introduced the concept of the dichromatic pattern (dcp), which is constructed using two interpenetrating lattices, ignoring any interfacial planes. The term dichromatic refers to the black and white dots conferred on the two lattices. The most symmetrical pattern is called the precursor (Fig. 7-2a). Any modification induced by the introduction of firstly an interface and secondly an interfacial defect leads at each step to a loss of symmetry elements, and consequently to the creation of crystallographic equivalent variants related by the lost symmetry operations. These variants may be found along the same interface. GB dislocations separating such energetically degenerated variants have been observed by Pond and Smith (1 976) and Bacmann et al. (1 98 1 ) in I=5 and 2=25 germanium bicrystals. The Burgers vector of these dislocations does not belong to the DSC lattice and as such is called a partial DSC dislocation.

387

Using group theory, these concepts have been extended to crystal coincidence by Gratias and Portier (1982) and Kalonji and Cahn (1982). In addition, these authors showed that symmetry constraints might explain some interfacial properties, such as the morphology of a crystal growing (during precipitation or solidification) in a crystalline environment. Pond (1989) reformulated this notion and introduced the dichromatic complex (dcc) which is the superposition of the two crystals with all their atoms (Fig. 7-2b). This allowed him to propose a unified description of all possible bulk and interfacial defects arising from the symmetry break of the bicrystal complex. A real bicrystallography has been defined to describe all the symmetries of an interface. Sagalowicz and Clark (1995, 1996) applied this bicrystallography to determine all the Burgers vectors that can be found in a I=13 GB.

a Figure 7-2. Projection on the (001) plane of a) dichromatic pattern (dcp) and b) the dichromatic complex (dcc) corresponding to [OOl] Z=5GB in a cubic diamond crystal. Circles refer to lattice sites of crystal I (black circles) and crystal I1 (white circles). Large circles are at the 0 level, and the small circles are at the % level along the [OOl] axis. Squares refer to the additional atomic sites of the lattice. Large squares are at level X and small circles are at level %. The two interpenetrating f.c.c. lattices make up the dcp which has a space group ( I 4/rn rn'rn') where '"" denotes an antisymmetry element, Le., a change in color. The two interpenetrating cubic diamond crystals make up the dcc which has a ( I 4,larn'd') space group. White and black dots refer to lattice sites I and 11, respectively. The square dots refer to the additional crystal sites.

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7.2.3 Dislocation Model Historically, the modeling of interfacial structure in terms of dislocations was developed before the geometrical concepts (Burgers, 1940; Bragg, 1940; Read and Shockley, 1950, 1952). The first experimental evidence of the validity of the dislocation model was given by Vogel et al. (1953). They were able to correlate the spacings of etch pits in low angle germanium GBs with the spacings of dislocations predicted by the Burgers model from the measured misorientations between the grains. The most direct evidence for the presence of dislocations in GBs came with the first observations by TEM of dislocations in aluminum (Hirsch et al.. 1956). The complete treatment was reviewed by Christian (1981), Forwood and Clarebrough (1991), and Sutton and Balluffi ( 1995).

7.2.3.1 Primary Dislocation Network Frank (1950) was the first to develop a theory that provides the dislocation density of a general GB. The demonstration was realized by making a Burgers circuit using the FS/RH (final stadright hand) convention with the closure failure being done in good crystal (Hirth and Lothe, 1968). Bilby et al. (19%) extended the formula to heterophase boundaries and demonstrated it using the concept of a continuous 2D distribution of dislocations at the interface. This concept of a continuous distribution of dislocations was independently defined in the 3D-case by Kroner ( 1981, which is an updated English version of a 1958 paper). We give here the Frank-Bilby formula reformulated i n matrix notation by Christian (198 1). The total Burgers vector S of the dislocations cut by an arbitrary vectorp of the GB is given by

s = (R

- R n1 p = p I -pII

(7-7)

where R , and R,, are the rotation matrices defining the orientation of the crystals I and I1 with respect to a reference lattice (where all the vectors and matrixes are expressed). S can be considered as the difference between the same vector p expressed once in the crystal I lattice base ( p l )and once in the crystal 11 lattice base (pII). If one of the two crystals, crystal I for instance, is chosen as the reference lattice, R, and R,, become respectively the identity Z and the matrix rotation R , and Eq. (7-7) simplifies to

S = (Z-R-I) p I

(7-8)

If p is now considered as an O-lattice element, this equation is identical to Eq. (7-5): S is consequently a lattice vector, which is always the case for a Burgers vector of a perfect dislocation. Thus Eq. (7-8) corresponds to the O-lattice treatment presented in Sec. 7.2.2: regions of best fit (O-elements) are separated by regions of worst fit, whose dislocation content is given by Eq. (7-8). This dislocation distribution is called the primary dislocation network, because the reference is a perfect lattice crystal. If the reference lattice is taken to be the median lattice, which is obtained from lattice I (respectively, 11) by a rotation Ron of axis e and angle 8/2 (respectively, R-elz), Eq. (7-7) simplifies to the original Frank’s formula

@- ~e p S = 2 sin 2 le1

(7-9)

Equation (7.9) is particularly interesting when p is a vector perpendicular to the rotation axis. In this case, the modulus of Eq. (7-9) gives

8 Ip I I S I = 2 sin 2

(7-10)

Equations (7-7), (7-8), (7-9), and (7-10) are geometrically valid for any kind of GB, but


their physical meanings were more satisfactory in the case of low angle GBs. Indeed, in the case of low-angle GBs, these formulas give a low density of dislocations. Dislocation cores do not overlap: individual dislocations can be clearly defined and, as we will see in the next part, the elastic theory can be simply applied to determine the stress field and Eq. (7-10) can be approximatedby l S l / l p l = 8 . However, in the case of high-angle GBs, as we will see in Sec. 7.2.4.3, these formulas can be very useful to determine the number of SU-dislocations that can be assembled in a GB model. One of the difficulties with these Frank-Bilby/O-lattice descriptions is that the relationship R between the crystal lattices is not unique: any operator U of the original crystal face group gives a new relationship U- ‘RU,which gives a different dislocation content. In practice, the relationship R is chosen in order to reproduce the experimental data. Rouvihre and Bourret (1990a) found that the rotation R leading to a minimum b2 value, where b is the total Burgers vector associated with a GB period, reproduced well the experimental observations of [OOl] tilt GBs.

7.2.3.2 Secondary Dislocation Network In high angle GBs, it is sometimes more interesting to consider the GB (rotation R ) as a deviation from a reference high coincidence GB (rotation R,). In this case, the reference lattice is not the perfect lattice crystal, but a lattice reflecting all the translations of the high-coincidence bicrystal, that is, the DSC-lattice of the high coincidence GB, which is denoted DSC, in Sec. 7.2.2. Bollmann (1970) showed that this deviation can be accommodated by a periodic network of dislocations, which are called sec-

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ondary dislocations, and whose distribution is superimposed on the coincident GB. Similarly to the primary dislocation case, the secondary dislocations are located in-between the 02-elements, which are determined from Eq. (7-6). The density of the secondary dislocations is also deduced from Eq. (7-6) by substitution of the period p of the GB in place of ro2:the obtained Burgers vector b, of the secondary dislocations is thus a DSC, vector. As the disorientation 68 (associated with 6R) is small, Eq. (7-10) (with 68instead of 8) can be used to estimate b,. The Burgers vector of these dislocations can also be determined by “circuit mapping’’, as defined by Pond (1989). A closed circuit is realized in the GB and is repeated in the DSC, lattice: the closure failure gives the Burgers vector. Thibault et al. (1994b) applied this method directly on HREM images and characterized the primary and secondary dislocation content of any part of a GB. Generally, these secondary grain boundary dislocations (GBDs) are linked to the GB and cannot move in one or the other adjoining grain. In addition, they are generally, but not necessarily, associated with a GB step. The height h of the step associated with one particular GBD is geometrically determined by the CSL. Determination of the Burgers vector can be done in a conventional manner using the two beams extinction condition in a TEM. If the dislocation density is too high or if the Burgers vector is too small, this method is not very sensitive. If HREM is applicable, the projection on the observation plane of the Burgers vector can be extracted and the perpendicular component can be obtained by symmetry considerations. An example of such a determination (Fig. 7-3) directly on an HREM image of a DSC dislocation in (122)(011),Z=9 is shown

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Figure 7-3. Determination of the (b, h ) pair directly from an HREM micrograph. The case presented here is the one o f a G B D i n (122) ( 0 0 1 ) Z = 9 . s l = 1/2 [OII l l and s 2 = 1/4 [21 Ill,; s2 is the projection of 1/2 [ I or 1/2 [IOl],, and 6 = s , - s 2 can be expressed simply in grain I as 1/6 [121], or 1/6 [ I 121,. h is the projection on the GB normal of (sI+s2)/2and is equal to the absolute value of 119 [ 1221, = a/3 (El Kajbaji and Thibault-Desseaux, 1988).

in El Kajbaji and Thibault-Desseaux (1988). The (b, h ) pair is determined using the method given by King and Smith (1980) in the following way: If s , and s2 are the displacements of the CSL due to the GB defect ( b , h ) respectively in grain I and grain 11, then b is given by the difference between s 1 and s 2 and h is the projection parallel to the GB plane of the half sum of s 1 ands2.

7.2.3.3 Stress Field Associated with Grain Boundaries Predicting the stress field of a GB is of prime importance, because the stress field can induce the attraction of impurities or lattice defects (point or linear defects) to the GB.

A general expression for the stress field of a low angle periodic tilt GB in an isotropic material was published by Hirth and Lothe in 1968 [Hirth and Lothe (1 982) is the new edition of this book]. The elastic stress field is calculated by adding the stress field of each individual dislocation of Burger vector b separated by a distance D. It is found that, with increasing distance x perpendicular to the GB, the stress field decays exponentially to zero over a distance comparable to the period D of the dislocation array. That is, contrary to an isolated dislocation, there are no long-range stresses in a periodic array of edge dislocations with b perpendicular to the GB. For instance, we give, respectively in Eq. (7-1 1) and (7-12), theexpressions of the shear component ax?of an edge dislocation array (for regions where x % D/2 x1) and of a single edge dislocation. The other stress components behave similarly. As usual in this section, y z is the GB plane and the dislocation lines are along z. The ,uand II are, respectively, the shear modulus and the Poisson’s ratio.

exp

(

--2

~ x ) c o s ( ~ ) (7-11) (7- 12)

From these two equations, it can then be seen that at a distance x = D 0,= 0), the stress field of the periodic GB dislocations is 0.07 times the stress field of the corresponding isolated dislocation. Near the GB (x3k2 (O'c 8'= 0 < 36.87")Domain: Intermediate GBs belonging to this domain can be separated into two families for disorientations close to 2 = 5 (e= 36.87'), such as the (1130) 2 = 6 5 (8=30.57") GB, the principle of continuity can be relatively well applied, although there is competition between a strict application of this principle (model {A++A+- A++pA+- A++A+- p}, noted Z3.3)and an adapted principle of continuity taking into account the A++A+- grouping 1, noted (model { A++A+-pA++A+-A++A+-p Z2J. For lower disorientations, this principle of continuity does not work: the core of the GB becomes more complex and has multiple structures. Forthe(510)Z=13 (8'=22.62')GB,two very different types of structure have been

Figure 7-8. 3D structure determination of the (310) [OOl] 2 = 5 tilt GB in germanium. a) a-fringe contrast showing areas of different translation states. A-D-G are dislocations (introducing no RBT), whereas BC-E-F are defects introducing a RBT (Bacmann et al., 1985). b) and c) are HREM pictures showing the projection of the GB along [OOl] and [130], respectively, and revealing a RBT along the tilt axis of about 0.073 nm (Bourret and Rouvikre, 1989).

41 0


simultaneously present in the same specimen. The first structure, which was called M, is strictly periodic and contains six SUs per period. Its period can be analyzed as { A++BU--B0+'AC-Bo'-BO3 1. In fact, this structure is only observed on a small scale. Moreover. under the electron beam it tends to be replaced by a varying structure (Rouviire and Bourret. 1989,1990b), which destroys the strict GB periodicity. Along the [OOl] direction, this varying structure has a rather symmetrical aspect. More precisely, the structure is composed of discrete dislocation cores well separated by perfect crystals. This second type of structure is the only type of structure observed for lower angle GBs such as the (610) .Z=37, (710) 2'=25, and (910) 2'=41 GBs. So at first sight. in this second type of structure, the dislocation cores seem to have a mirror plane symmetry identical to the model S of the (310) 2'=5 GB. However, more careful analysis shows that the core structure is extremely variable. In fact, it is found that many atomic models, which have a high number of SUs, can be constructed and simply related to each other: without difficulty, it is possible to construct tetracoordinated models which are not periodic along the [OOl] tilt axis. These models can be analyzed as a superposition along [OOl] of periods of different periodic models. In the case of the (5 10) 2'=13 GB, mainly only two asymmetrical periodic models, IH+ and IH-, are involved i n these complex stackings, (Rouvikre and Bourret, 1989).The IH' and IH- models are symmetrically related (this is the meaning of the "c" and '*-" superscripts here). Their period is composed of the same SUs. i.e., {ppA++B 0-- B +& } , In summary, in this second type of GB structure, not only can the GB structure vary from one period p to the neighboring one, but it can also vary along the tilt axis. Charai et al. ( 1 998 a) confirmed that the number of ba-

sic periodic models in the (710) 2 = 2 5 were more numerous. Using Z-contrast scanning TEM, Chisholm et al. (1999) clearly identified the core of a complex tetracoordinated periodic model ((pA+-A-+A+-A++))for this same (710) Z = 2 5 GB. They found that the electron beam produces a perturbation of the structure, which becomes more symmetrical and most likely contains vacancies. The fact that the 2D-lattice of the GB is centered ( a = l ) , as in the case of (610) Z = 3 7 ( 8= 18.92') does not modify the above tendencies (Ruvimov et al., 1994). Sagalowicz and Clark ( I 995) calculated all the possible secondary dislocation networks of a (510) 2 = 1 3 GB and observed imperfect dislocations of Burgers vector 1/4 [OOI], which link two variants of the GB. They measured a RBT of about 1/4 [OOl]. In general, the static GB energies of the basic periodic models involving more SUs than the symmetrical (S)- and zig-zag ( Z ) type modes constructed using the principle of continuity, have slightly higher energies than those of these ( S ) - and (Z)-type models. However, the configuration entropy term is in favor of stabilization of the complex structure involving superimposition of periods of periodic models. Hardouin Duparc and Torrent (1996) show that unlike the .Z= 1 1 (01 1) tilt GB, 2 = 2 5 starts disordering several hundreds of Kelvin below the silicon melting temperature, which is consistent with the larger bond and angular disorder in (01 1) tilt GBs (Ihlal et al., 1996a). The authors showed that an extension of the disorder parallel to the GB plane is almost the same in (011) and (001) GBs (about 0.3-0.4 nm), but the strain field is larger in (001) tilt boundaries. The consequences on the electrical properties and the segregation behavior will be discussed in the following sections.

7.3 Grain Boundary Structure: Experience and Simulation Results

The k1 3"). In carefully controlled germanium bicrystals (170) 2 = 2 5 , (Charai et al., 1998a, b), sulfur segregates at particular sites in the primary dislocation cores. X-ray dispersive spectroscopy clearly shows a sulfur enhancement at the GB plane. The core structure of the GB itself is changed, as evidenced by HREM, and the energy filtered image at the L2,3sulfur edge shows that the specific sites for segregation are within the primary dislocation cores.

7.5.4 Transition Elements It was recognized very early that most of the electrical activity of GBs after annealing was a consequence of impurity precipitation. In that respect, 3d transition metallic impurities, whether introduced deliberately or not, are especially active. As a consequence, numerous works were carried out most of them induced by undesirable contamination and precipitation occurring during annealing. For instance, on as-grown (510) 2'= 13 and (7 10)Z= 25 silicon GBs Ihlal and Nouet (1989) measured no EBIC contrast. After annealing between 750 and 950°C strong uniform or dotted EBIC contrasts appear. This has also been observed by Maurice and Colliex (1989) on a (710) 2 = 2 5 GB in silicon, as well as on a Z= 3 coherent and incoherent twin (Fig. 7.15). Colonies of pre-

433

cipitates are at the source of such electrical activity and the segregation of impurities is itself dependent on the defect content of the GBs. For example, recombination is larger in an asymmetric plane or when the disorientation from a twin orientation increases. Finally, several authors (Broniatowski, 1989) have noted the effect of the cooling rate. At a high cooling rate, the activity enhancement is by far greater than after slow cooling. This points out the importance of segregated impurities on electrical activity. Copper, nickel, and iron were among the most studied. It should be noted that few calculated electron densities close to a transition metal at a GB are available. Up to now, chromium (Masuda-Jindo, 1994) and nickel (Torrent, 1996) are the only ones studied in detail.

7.5.4.1 Copper Copper is a very fast diffuser in silicon, and has been known for a long time to precipitate at defects, such as dislocations, stacking faults, or GBs. Contamination by copper is very easy and introduces unwanted electrical activity. After 10 min at 900 "C, the GB is active and has adensity of states close to 2 x 10l6eV m-2 over a broad energy range inside the gap (Broniatowski, 1987). With a longer annealing time, a well defined DLTS peak is observed at 0.53 eV and is attributed to copper precipitation at the GB due to pollution during the annealing process. The copper decorated Z= 25 GB was characterized by TEM (Broniatowski, 1989). It contained microprecipitates, generally in colonies with an average density of l O I 4 m-2. Broniatowski (1989) interpreted the spectra by introducing a Schottky barrier at each semiconductor-recipitate interface. Thus each precipitate acts as a multiply charged trap

434


and is fundamentally different in nature from the traps considered so far. When copper is deliberately introduced in quantities larger than the solubility limit (Maurice, 1993; Ihlal et al., 1996a), precipitation occurs both in the bulk and at the GB. A denuded zone is formed on both sides of the GB, the width of which is very dependent on the gettering efficiency of the considered GB. The most active is 2'=25, fol13, and 2'= 9. This order followed by I= lows the energy per unit interface area of these particular GBs. The exact morphology and structure of the colonies of precipitates has been studied in detail by Broniatowski and Haut (1990) and El Kajbaji et al. (1992). The precipitates are in an epitaxial relationship in a 2=25 GB and the CuSi phase is close to the 0-phase. a b.c.c. structure. The central part of the colonies seems to nucleate at additional dislocations in the GB plane. Their formation induces a large volume variation which should be accommodated by silicon interstitial emission. The GB is an easy site for emission or absorption of these interstitials, explaining the development of colonies in the GB plane. 7.5.4.2 Nickel

Nickel is, like copper, a fast diffuser in silicon. It also introduces electrical activity and precipitates preferentially at GBs after rapid cooling (Broniatowski and Haut, 1990: Ihlal et a]., 1995: Portier and Rizk, 1996: Portier et al., 1995). However, due to a small misfit with silicon (0.4'%), the precipitates are in the form of disilicide platelets, Nisi,, with a fluorite structure similar to the bulk precipitates. In 2'=25 GBs. epitaxial growth on the ( 1 1 1 ) silicon planes is observed in one of the grains with or without an associated twin. These precipitates are visible by EBIC and the

EBIC contrast is a function of the size and distribution of Nisi, precipitates present in the GB. The electronic structure of individual nickel impurities located in the GB plane of a Z= 25 and a 2'= 13 was studied by Torrent ( 1996). He demonstrated that the segregation energy was maximum for the substitutional position as opposed to the interstitial one. This energy is -2.6 eV in I = 2 5 and - 1.3 eV in .E'= 13, in good agreement with experimental results. In substitutional sites, the nickel atoms have a semi-metallic behavior with partially occupied states in the gap, as observed. In addition, there is a large transfer of charge with the neighboring silicon atoms. As a result, a fully consistent picture of the nickel segregation has emerged. It should be noted that an earlier study on chromium (Masuda-Jindo, 1989) has drawn similar conclusions. 7.5.4.3 Iron

Iron is known as a moderately diffusing element in silicon with a low solubility in comparison with nickel or copper. As a consequence segregation and/or precipitation at GBs is not observed, unless high temperature treatments are performed (< 1200°C) (Portier et al., 1997; Ihlal et a]., 1996b). Contrary to the case for copper or nickel, a slow cooling rate induces the formation of iron silicides at GBs, whereas segregation with no visible precipitates occurs at a high cooling rate. Iron silicides exist in at least five different phases in bulk silicon. Both &-phase FeSi and a-phase FeSi, are observed in Z=25 GBs. GBs containing precipitates are electrically active, as evidenced by EBIC and DLTS. The energy levels in the gaps are between -0.48 and -0.59 eV from the conduction band.

7.6 Mechanical Properties of Grain Boundaries in Semiconductors

7.5.5 Conclusions

It should be pointed out that the segregation of a combination of impurities is also possible. Following the scheme responsible for the gettering of metallic impurities by oxygen precipitates (Cerofolini and Maeda, 1989), the process could be described by two steps in the following way:

- The GB secondary dislocations or extrinsic dislocations attract the interstitial oxygen in their elastic field. The most active dislocations will be those with the largest edge component. - The oxygen-rich region attracts metallic impurities giving metal-rich particles at the GB; these are particularly electrically active. Oxygen gettering is generally optimized between 650 and 9OO0C, which is the temperature range in which GBs are very active. Therefore the most active GBs would either be those of low surface energy ( 2 = 9 ) , but containing the largest amount of dislocations or steps associated with a dislocation, or the ones with the highest interfacial energy (like Z'=25) and containing a lot of segregation sites. As a conclusion, the role of impurities in GBs is very important. Unless very special care is taken, bicrystals containing a single GB are the most likely areas of impurity segregation after annealing treatment. In polycrystalline materials, especially solar cells, each specific case should be examined. Most of the industrial materials contain a mixture of dopant, oxygen, and metallic impurities, not to mention hydrogen. All of these impurities are able to segregate or cosegregate at GBs with an overall effect on the electrical activity that is difficult to predict or even to analyze.

435

7.6 Mechanical Properties of Grain Boundaries in Semiconductors 7.6.1 Introduction The deformation of polycrystalline materials has been widely and initially studied for metals, alloys, and ceramics. The goal was to understand the behavior of materials under different temperature and stress conditions in order to achieve special properties. The role of GBs has been emphasized for a long time. The GBs were considered as a barrier to crystal slip and, as a consequence, they can harden the material; the grain size dependence of the yield stress led to numerous studies. At high temperatures, GB behavior was explained by diffusion along the interface, and the recrystallization or superplasticity of polycrystalline materials after high temperature deformation was viewed as the consequence of matter transport in both grains and at the GB. The role of impurity segregation in GB embrittlement is also a well-known phenomenon, and led to equally numerous studies in either polycrystals or bicrystals. Theoretical models were elaborated to explain the variable influence of different impurities on GB embrittlement. Although the literature on the role of GBs in relation to the mechanical properties of materials generally refers to metallic or ceramic materials, this is often a good starting point from which to approach the semiconductors case. In 1972, Hirth gave a review paper on the influence of GBs on the mechanical properties of metallic materials. In 1988, the concept of GB design was introduced by Watanabe, who emphasized the role of GB character distribution of polycrystalline materials and its link to both the bulk properties and to intrinsic GB properties.

436


In the case of semiconductor materials, this question of GB mechanical behavior arises during the processing, for instance, of polysilicon ribbon, which is used for low resistivity wires or as isolating material if suitably doped. Nanocrystalline materials grown artificially also contain numerous defects. Performance depends strongly on the stability of the material, and namely on the induced defects, their density, their mobility, and the interaction between different more or less welcome defects. Dopant impurities are welcome defects, and their spatial distribution has to remain under control during the process, Le., not perturbed by mobile dislocations or migrating GBs. Furthermore, large scale integration leads to specific problems related to the plastic properties of the material. Generally, it induces local stresses which are released by the emission of dislocations whose distribution evolves under further treatments. Furthermore, during recrystallization at high temperature, migrating GBs sweep and absorb dislocations whilst the size of the grains increases. These structural changes may induce unexpected properties and must be understood and if possible controlled. Several techniques are now available: in situ experiments using synchrotron X-ray topography or electron microscopy, and the post-mortem experiments using conventional and high resolution electron microscopies. As a consequence, the understanding of some special GB deformations has been clarified, and the very early stages of the interaction mechanisms between deformation-induced dislocations and GB are now better characterized. The generalization remains unclear. Furthermore, some numerical calculations are now able to give some indications on GB sliding or fracture.

7.6.2 Interaction Between Dislocations and Grain Boundaries Under applied or internal stresses and at a suitable temperature, dislocations are created in the bulk and then move within the crystalline material. Consequently, they react with all the defects in their path, and in particular with grain boundaries. The presence of a GB on the dislocation’s path generally blocks the slip, and dislocation pile-ups are formed in the vicinity of the GB. When two slip systems are activated, dislocations on one system can be stopped at the GB and may then be intercepted by the second slip system leading to locks. As a consequence, dislocation crossslipping is highly probable at the GB. As discussed by George et al. (1989), the neighborhood of a GB is in an advanced state of deformation. Furthermore, the vicinity of the GB was shown by HREM to be a preferential area for strong interaction between stopped dislocations and point defects created by the deformation (Thibault-Desseaux and Putaux, 1989). Smith (1982) gave a good summary of the different steps of the mechanisms occurring at a GB and George ( 1988) gave a review of the hardening mechanisms at grain boundaries in silicon as viewed from the microscopic point of view. Two major mechanisms could be proposed for stress release at the head of the pile-up, namely i ) the entrance and dissociation of the dislocations within the GB or ii) their transmission across the GB. As the reaction occurs, two conservation rules must be obeyed, firstly, that the Burgers vector is conserved and secondly that the step height associated with the GBD is also conserved. 7.6.2.1 Dislocation Absorption

The dislocations created by deformation are known to be stopped and absorbed by


GBs. Thus in the case of complete absorption, the rule is given by

(bo,ho) = C (bii hi) 1

where i refers to the ith GBD that the incoming dislocation is decomposed into. The entrance of the incoming dislocation as a whole would have created a global step whose height h, is the projection on the GB normal of half the Burgers vectorb, (by convention, the reference plane is the mean GB plane, see Sec. 7.2). Moreover, it has to be kept in mind that the height of the step associated with a given GBD Burgers vector can only take welldefined values determined by the DSC lattice geometry. The experimental determination of this was explained in Sec. 7.2. The appearance of DSC dislocations associated with a step can affect the decomposition of the incoming dislocation. Forwood and Clarebrough (1 981) predicted that nonprimitive DSC dislocations with no step would be stable in symmetrical tilt GBs despite the unfavorable b2 energetic criterion. The application of this rule is simple in relatively low 2GBs. In fact, this rule is geometrically necessary, but it is not so straightforward in polycrystalline materials where the GBs are experimentally difficult to characterize. Experimental evidence for the disappearance of the contrast of dislocations observed by TEM (Pumphrey et a]., 1977) suggested to Gleiter (1977) that in the case of general GBs, the cores of external dislocations spread out into dislocations with infinitesimal Burgers vectors. However, Pond and Smith (1977) and Dingley and Pond (1 979) pointed out that dislocations can enter both coincidence and noncoincidence GBs, where they dissociate into components belonging to the nearest DSC lattice. The rule of absorption and decomposition in DSC dislocations was experimentally

437

confirmed for the first time using conventional TEM by Bollmann et a]. (1972) in a metallic GB, and was confirmed using HREM by El Kajbaji and Thibault-Desseaux (1 988) and Skroztky et al. (1 987), respectively in silicon and germanium bicrystals, and more recently by Thibault et al. (1994a) and Michaud et al. (1993), who studied the deformation of C= 19 and C=5 1 germanium and silicon bicrystals. All the observed GBs were relatively well-defined; even for such high 2 values as 337, it was possible (Putaux, 1991) to detect individual GBDs with localized cores, unlike in the Gleiter model. Then absorption rule assumes that dislocations integrate the GB as a whole. However, the deformation induced dislocations 1/2(011) are known to be dissociated into two Shockley partial dislocations 1/6 (1 12) separated by an intrinsic stacking fault. Thus the question arises of how to overcome the repulsive interaction between the partials at the entrance event (King and Chen, 1984). This was clarified in the simple case of deformation of the 2=9 silicon bicrystal (Thibault et al. 1989) and confirmed in more complex cases (Thibault et al., 1994a). It turns out that the stress at the leading partial is released by the decomposition and emission of glissile GBDs into the interface. As a consequence, this occurrence leads to a block of the slip at the GB. On the other hand, with complete integration and decomposition of both the leading and the trailing partials in GBDs, the incoming dislocation generally requires simultaneous glide and climb of the GBDs. Figure 7-17 shows the integration of a dissociated 60" dislocation a/2(101)withinthe(122)(011)2=9 GBin silicon. In this case, the leading partial decomposed first by glide into the GB. One glissile DSC dislocation is ejected in the boundary. After the entrance of the trailing partial, the residual dislocation decomposed

438


Figure 7-17. a) Entrance of a 60" dissociated bulk dislocation in a (122) [ O l l ] Z=9 tilt GB in silicon. b) shows the decomposition of the leading partial in two GB defects. The three DSC dislocations result from the final decomposition: b,=a/18 I4111 is glissile in the GB plane, b,=a/9 [ 1221 is sessile, and bi0 is a mixed GB dislocation and carries the screw component. c) showsb, andb;,. The bond reconstructions along the [ O I I ] direction are the same as in the bulk. [HREM images taken from El Kajbaji and ThibaultDesseaux (19881.1

by climb (and glide) into two DSC dislocations called 6 , and b i o on the figure. This has been observed by experiment to take place at temperatures above 0.5 T,. Furthermore, at temperatures at which large

climb is possible, the dislocations common to both grains can be stopped even at the interface by climb decomposition (Thibault et al., 1989). In the case of highly anisotropic GBs, image stresses have to be considered. The image force can make matrix dislocations which are attracted to or repelled from the interface. This helps to pin the dislocations at the GB or promote cross-slip in the vicinity of the boundary (Khalfallah et al., 1990). It has to be pointed out that the core of the DSC dislocations stemming from the entrance of an incoming dislocation is not necessarily well reconstructed. It has been shown (Fig. 7-17) that an in-running 60" dislocation, dissociated into 30" and 90" partials, decomposes on entering into a 2 = 9 (01 1 ) tilt GB into three DSC dislocations, b30 being the screw component. The structure of the GBD core is not far from that of the bulk dislocation core, and can be described with the SU model using five, six, and seven atom rings as i n the bulk. In particular, the SU linked to be b, dislocation is the T SU. The problems linked with bond reconstruction along the GBD line are similar to those occurring in the bulk (ThibaultDesseaux and Putaux, 1989). The entrance and decomposition of deformation induced dislocations has also been studied in detail both by in situ 1 MeV TEM and X-ray topography in Z= 19, Z= 5 1, and Z = 9 germanium and silicon (01 1) tilt bicrystals, and also be HREM (Michaud et al., 1993; Thibault et al., 1994a) of 2=19 and Z=5 1. The dislocations are stopped even in low angle boundaries, and the structure of residual grain boundary dislocations has been determined by HREM. These studies showed that the intermediate stages of incorporation depend on the external conditions, Le., the type of the leading partial: 30" or 90" Shockley partial, and might lead to GBDs with a large core. These studies con-


firmed what has been already observed by HREM in the special case of 2=9 GB deformation (Thibault et al., 1989). Sagalowicz and Clark (1995 b) observed and studied in detail by TEM the entrance of lattice dislocations produced by compression into a [OOl] tilt Z'=13 GB. In this boundary, decomposition of the in-coming lattice dislocation is found to be more complex, and they found imperfect and partial dislocations. Imperfect dislocations are dislocations that separate two structures of a GB related by a symmetry element (thus having the same energy), whereas partial dislocations separate two GB structures corresponding to two different dichromatic complexes (see Sec. 7.2.2). In fact, imperfect dislocations were found to exist before deformation: the authors concluded that decomposition of the lattice dislocations induced by deformation may very likely result in partial dislocations whose Burgers vector is the sum of a 1/4 (1 1 1) plus a DSC vector. The role of the step height in the decomposition has also been emphasized. To sustain these results, the authors argue that in the case of [OOl] tilt GBs the differences in energy between different structures corresponding to the two different dichromatic complexes might not be so large. Thus the occurrence of these partial dislocations might be the consequence of a favored energy balance between the GB and the step energies and the elastic strain energy gain.

7.6.2.2 Dislocation Transmission Across Grain Boundaries The rule to be applied in the case of the direct transmission of a dislocation from one grain to the adjoining one is expressed by

(b1, hI)=(bII,

h I I ) + ( b g b , hgb)

where b&, hgb is the residue left in the GB whatever it may be and gb stands for grain

439

boundary. In the following, b, refers to a glissile GB dislocation and g stands for glissile. The transmission of dislocations occurs easily if there is no residual GB dislocation left after the transmission, i.e., when b g b is zero, otherwise the energetic balance may be unfavorable. In addition, the incoming dislocation line generally has to rotate in order to achieve a suitable slip plane in the second grain. This process involves climb into the boundary and is diffusion-limited. Forwood and Clarebrough (1981) provided evidence of direct transmission events across a GB (in stainless steel) containing a line common to the slip planes in both grains. In this case, incursion across the boundary is conservative and is only limited by the residue left in the interface. In-situ 1 MeV and X-ray synchrotron topography showed numerous direct dislocation crossing events common to both grains in well defined germanium tilt bicrystals whose tilt axis was in zone with slip planes in both grains (Jacques et al., 1987; Baillin et al., 1987). This process is likely to occur owing to the absence of a GB residue. By the same method, Martinez-Hernandez et al. (1 986) showed that the transmission of dislocations across the GB is more difficult for (001) 2 = 2 5 than for (01 1) 2 = 9 , because in the Z'= 25 case no slip system is common to both grains in contrast to the 2 = 9 case. Nevertheless, the passage across the boundary of a dislocation belonging to an uncommon slip system was observed during 1 MeV in situ experiments in germanium and silicon bicrystals provided that sufficient stress concentrations arise at pile-up tips (Baillin et al., 1990; Jacques et al., 1990). Observations taken of dislocations emitted from an impact point corresponding to a pile-up are in favor of an indirect process because of the numerous dislocations accumulating in the same area. Furthermore, as pointed out by

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Michaud et al. (1993), the influence of the decomposition of the incoming dislocation as well as the influence of the nature of the heading partial is of prime importance in the dislocation transmission across the GB.

7.6.2.3 Grain Boundaries as a Dislocation Source The GBs are a major source of bulk dislocations as well as GBDs during the deformation of polycrystalline materials, although the nature of the dislocation sources is still far from being clear. Nevertheless, since it strongly influences the subsequent evolution of the material under further treatments, it is of fundamental importance to attempt to get a better understanding of the process. The direct emission of a dislocation from a GB in a perfect coincidence position seems to be energetically prohibitive. However, in the case of a GB already containing defects, this phenomenon is more likely. Two kinds of dislocation can be emitted from the GB into the matrix. The first one is a perfect lattice dislocation, mainly 112 (1 lo), as discussed by Baillin et al. (1990), the second being a partial lattice dislocation 1/6 (1 12), which remains linked by a stacking fault to the boundary (Fig. 7 - 18). Both conventional TEM and HREM (George et al., 1989; Jacques et al. 1990) also revealed this second mechanism. It must be noted that conventional TEM shows the formation of extrinsic stacking faults, which are in fact the superposition of distinct intrinsic stacking faults emerging from the GB on nonadjacent ( 1 1 1 ) planes. Being close to the GB, this leads to complex configurations which will contribute to locally harden the material and which have a strong influence on the response of the material to chemical or thermal constraints via the presence of a high density of defects. Thus the emission of dis-

Figure 7-18. Dislocation emission from a GB. TEM image a ) shows the emission of a pair of Shockley partials from a Z= 9 GB in germanium. The weak beam micrograph b) shows the two partials. The contrast between the fringes is due to an extrinsic stacking fault, each partial having trailed its own intrinsic stacking fault [George et al. (1989)l.

locations from the GB may be determined by the GB defects already existing within the interface.

7.6.2.4 Grain Boundary Dislocation Movement GBDs can move along the boundary as lattice dislocations in the bulk, that is, by


glide and by climb. GBDs with Burgers vectors parallel to the GB plane can glide in the interface, whereas those with Burgers vectors perpendicular to the GB plane move by climb. If the GBD is associated with a step, then the GB migrates laterally as the GBD moves along the interface. If the GBD’s Burgers vector has a component parallel to the interface, then the GBD motion results in GB sliding. The mobility of a glissile GBD has been studied by HREM and X-ray topography (Benhormaet al., 1991; Jacques et al., 1993). The authors found a strong anisotropy: the edge segments were stiff and slow whereas the screw segments were wavy and rapid. This anisotropy was explained in terms of GBD and kink core structure. While moving along the GB plane, a GBD can interact with all the GB defects, such as secondary dislocations, steps, or precipitates. All these obstacles are part of the resistance of a GB against the entrance and decomposition of lattice dislocations. This can lead to GBD pile-ups which in turn can leave residual stresses stemming from the lattice and the GB. As Hirth (1972) mentioned, if the resistance is low, complete recovery of the GB can be achieved. GBDs are free to rearrange in order to give a new GB structure without long range stresses. This has been confirmed by HREM observations (Putaux and Thibault, 1990) and details are given in Sec. 7.6.3.2.

7.6.3 Physical Consequences 7.6.3.1 Grain Boundary Migration It is well known that GB migration controls the grain size and orientation as well as impurities’ redistribution. In this sense, it plays an important role in the overall properties of the materials and has been extensively studied in metals and ceramics. In the case of semiconductor devices, the occurrence of GBs and their migration might lead

44 1

to unexpected effects, which have to be known even if they are not well understood. The phenomenological equation giving the velocity of the GB is at least in the Newtonian regime

v=Mf where M is the mobility of the GB andfthe driving force. The mobility is controlled by a wide range of intrinsic or extrinsic obstacles and it is generally reliant on the temperature. The force can either an externally applied stress or an internal driving force stemming from the free energy difference of the two grains. Diffusivity across the GB is one of the thermally activated processes which accounts for the GB’s mobility. The impurity content influences the GB velocity, which can exhibit nonlinear behavior as a function of the driving force; at high velocity, however, the impurities no longer have an effect [see the review paper by Bauer (1982)l. Diffusion-induced grain boundary migration (DIGM) is a phenomenon observed in some systems at temperatures where the bulk diffusion is extremely slow. The diffusion of solute atoms into the bulk or from the bulk via the GB result respectively in an alloyed or a de-alloyed region behind the GB path. Two different mechanisms were proposed. Hillert and Purdy (1 978) invoked a chemical potential gradient, whereas (Balluffi and Cahn (1981) invoked GBD climb. In the first case, a chemical potential gradient would lead to an asymmetrical strain which would be relieved by GB migration, leaving an alloyed region behind the GB. In the second case, the GB would migrate through the movement of GBDs associated with steps; the climbing driving force would come from the supersaturation of point defects arising from a GB Kirkendall effect due to there being no net fluxes of solutes and/or solvent along the GB. The first mod-

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el is based on the anisotropical response of two continuous adjoining grains, whereas in the second model an attempt is made to account for the structure of the GB. From TEM observation of DIGM in a NiCu couple, Liu et al. (1988) argued in favor of the coherency strain model of Hillert and Purdy (1 978) A dislocation wall at the original GB position is compensating the misfit between the matrix and the alloyed region, and the movement of the GB in both the forward and backward direction cannot be explained by the dislocation climb model. Vaudin et al. ( 1988) supported the same idea using the strong evidence of GB migration in MgO by diffusion of NiO along the interfaces. DIGM was studied as a function of rotation angle in a series of symmetrical tilt GBs by Chen and King (1988) and in a series of asymmetrical tilt GBs by King and Dixit (1990). The response of the GBs to DIGM differs strongly whether they are symmetrical or not. The authors pointed out that apparently contradictory results could be explained if it is considered that the coherency strain provides the driving force, whereas GBD climb is one of the migration mechanisms (occurring exclusively in some cases), with other mechanisms such as atom rearrangement across the interface taking place as well. During the processing of polycrystalline semiconductors, these considerations must be taken into account even if, compared to the situation in metals, the diffusion and migration processes are slow in covalent materials. Smith and Grovernor (1986) observed the enhancement of boundary mobility in silicon and germanium films by doping, even at high temperature. In fact, the GB can migrate under an applied stress or a differential chemical potential. The mechanisms taking place, however, could be of two types, namely i) the col-

lective but local motion of the GB primary dislocations or the extension of pure steps perpendicular to the GB plane; this can be viewed as local atomic rearrangements across the boundary in the case of pure steps movement, or where the primary dislocations could glide laterally. Or ii) a displacement of individual GBDs (associated with a GB step) over large distances along the interface. Both mechanisms require glide and/or climb, depending on the Burgers vector or more precisely on the resultant of the Burgers vector involved in the processes. On the one hand, local and collective glide of GBDs can lead to high energy GB configurations, whilst on the other hand the climb of a GBD over large distances requires high diffusion conditions. The migration of a pure GB step by local rearrangement of the atoms at the GB was observed by HREM in situ heating experiments (Ichinose and Ishida, 1990). The authors were able to observe both the migration of a (1 1 1) ,Z= 3 facet in a (1 12) E= 3 GB in silicon and the subsequent structural changes. They showed that through a local rearrangement of the atoms, the facet migrates perpendicular to the (1 l l ) plane by steps of the size of the CSL unit cell. This mechanism avoids a rigid translation of one grain relative to the other, which would result from the glide of a/6(112) GBDs on the (1 1 1) plane. However, the portions of the migrating GB are small (1 11) facets of the E = 3 GB connected by ( 1 12) facets, and this provides limiting conditions which might impose on the migration mechanism. 7.6.3.2 Recovery of the Grain Boundary Structure and Cavitation

If the number of extrinsic dislocations increases in a GB, the long-range stresses also increase. As Valiev et al. (1983) and Grabski (1985) recalled, the complete re-


covery of a GB occurs by motion, by annihilation of GBDs and their decomposition into smaller GBDs, and by rejection or absorption of lattice dislocations in order to obtain a GB in a new equilibrium configuration without long range stresses. The recovery may be accompanied by the formation of subgrain boundaries within the grains. The recovery of germanium bicrystals by the rotation of two adjacent grains was observed macroscopically by Bacmann et al. (1 982a). In addition, HREM has allowed the study of the atomic structure evolution undertaken by a (1 12) 2=9 silicon tilt bicrystal during deformation (Putaux and Thibault, 1990). The original rotation angle was found to change from 38.94' to about 56" in compression experiments. The accumulation of deformation induced dislocations and their decomposition within the boundary has led to the observation of different 2 GBs sharing the same (011) mean plane, such as (599) 2 = 1 8 7 (8=42.9"), (233) 2 = 1 1 (8=50.5"),and (344) 2 = 4 1 (8=55.9"), all of whose structures have been clearly identified, Furthermore, the formation of subgrain boundaries in the grains has been clearly observed. The junction point between the subgrain boundary and the highangle boundary delimits two parts of the high-angle GB with two different structures corresponding to two slightly different rotation angles (Thibault et al., 1991). The recovery of the GB structure taking place after low strain is made by the accumulation and homogeneous redistribution of one particular structural unit called T (Sec. 7.3) linked to the 2 = 9 primitive DSC dislocation a/9 [122]. However, after high strain, the recovery leads to the formation of more complex structures, which involve new structural units. The interaction between GB dislocations and the further evolution of a GB dislocation network has been also ob-

443

served by TEM in a deformed 2=13 [001 ] tilt silicon GB (Sagalowicz and Clark, 1995b) as the compressive stress increases. As mentioned in Sec. 7.6.2.1, at high temperatures, slip transfer across the GB becomes unlikely and most lattice dislocations are trapped within the GB where they lower their energy by decomposition into smaller dislocations. These GBDs can then move easily by glide and climb along the boundary. As in the crystal, they can form pile-ups in the interface at triple junctions between GBs, or at precipitates, and this promotes conditions for cavitation. Lim (1987, 1988) produced a good review on the problems connected with GB cavitation occurring at high temperatures. The grain boundary fracture of a Z=25 bicrystal of silicon has been investigated by Jacques and Roberts ( 1996). The influence of the applied stress and the temperature have been studied, and it was shown that i) fracture only starts at dislocation pile-up, and ii) fracture is delayed when dislocation transmission across the GB has time to occur.

7.6.4 Deformation Modelling As observed in former paragraphs, the deformation mechanisms lead to complex configurations either in the grains or at the GB. Potentially, modelization could provide a better understanding and a more accurate prediction of the evolution of a material under stress. A first attempt to achieve modelization of a dislocation microstructure has been made by Kubin and Canova (1989). They were able to compute the 3D dislocation distribution and its evolution under a given applied stress. A recent review has been given by Devincre and Kubin (1997). The application of such a modelization to polycrystals would open the way to better predictions of their mechanical properties.

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The atomistic simulations are far less developed than in other domains. Nevertheless, some attempts have been made to compare experimental results and simulations [see, for instance, examples in ordered metallic compounds by Pestmann et al. (199 1 )]. The main point remains to know the intrinsic elastic properties of the GB, i.e., the tensor of the elastic moduli. Alber et al. (1992) made an attempt to define local elastic moduli for a GB considered as a heterogeneous continuum made of atomic polyhedra associated with the atomic level moduli calculated at the atomic positions. Using Tersoff’s potential. Marinopoulos et a]. ( 1 998) extended this idea to covalent materials namely (122) [Ol 11 2 = 3 tilt GBs in silicon. The elastic moduli associated with atoms at the interface are found to vary strongly: the decrease in the local coordination has a strong effect on the elastic moduli tensor, which was shown to not be positively definite in the case of a GB model with threefold coordinated atoms. Recently. atomistic simulations have been performed on GB sliding and fracture. The 2 = 5 twist GB sliding has been simulated by Molteni et al. (1996). During the sliding process, the energy of the GB oscillates, strongly indicating that local rebonding takes place for some particular translation states. The electronic density of states is also varying: at all energy jumps the number of states in the gap is reduced or eliminated. At a large strain, the disorder extends into the adjacent grains and produces weaker bonds, which would lead to fracture. At a finite temperature, the oscillation amplitude decreases, indicating that the sliding process might be facilitated. Kohyama ( 1999) performed ab initio tensile tests of GBs in a nonpolar (122) [Ol I ] 2 = 9 in S i c and showed the successive bond breaking. The first to break is a C-C bond next to the interfacial five-atom ring, followed by the

two Si-C interfacial bonds. As mentioned by the authors, fracture occurs for a stress larger than the experimental one, because the simulated system contains no defect other than the interface.

7.7 Conclusions Grain boundaries in semiconductors are 2D defects which strongly influence the overall properties of materials through their response to external and internal constraints. However, they are not the only defects present in the materials. Consequently, if the external conditions are changing, strong interaction results between the different types of defects, which may lead to large modifications of the material properties. The ultimate purpose of device design would be to keep these interactions under control, but this implicitly first requires a perfect knowledge of the structure and its mechanisms. As this chapter has shown, the geometrical description is now well established and provides strong support for the corresponding energetic descriptions. Simulations can describe the atomic and electronic structure of some special GBs containing a few hundred of atoms. Nevertheless, owing to increasing computer power, they are on the way to predicting the structural multiplicity of simple GBs as well as the structure of more general GBs. The atomic GB structure can be solved experimentally in the favorable cases of GBs with a low index tilt axis. Most of the studied GB structures were compatible with structures with bonds completely reconstructed and, furthermore, it was found that the structure of GB defects obeyed the same constraints. A determination of the atomic structure of more general GBs has not up to now been achieved, despite the existence of

7.8 References

certain investigation means, such as STM or FIM, which could provide chemical characterization but are still not yet used extensively. We are still, moreover, far from knowing all there is to know about GB interactions with other defects such as point defects, impurities, or dislocations, although new results have appeared during the last ten years. In addition to this, the correlation between nanoscopic level mechanisms and macroscopic level properties has not been clearly solved either yet. Nevertheless, the main point to emerge is that the electrical activity of GBs does not stem intrinsically from the GB itself, but comes essentially from impurities present at the GB which are attracted by the stress field of GBDs. In fact, apart from some low-energy GBs, which are in their energy ground state only under special limiting conditions (for instance, a bicrystal without any stress), most of the GBs encountered either contain linear defects or absorb them. The complex reactions between defects occurring at the GB (in the interface and in the vicinity) may involve more or less long-range stress fields which will have an influence on the properties of the material.

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Szkielko, W., Petermann, G. ( 1 9 8 3 , in: Poly-microcnstalline and Amorphous Semiconductors: Les Editions de Physique, Les Ulis, France, pp. 379385. Tafto, J. (1979), in: Proc. 39th Annual Meeting EMSA, San Antonio: p. 154. Takai, T., Choi, D., Thathachari, Y., Halicioglu, T., Tiller, W. A. ( 1 9 9 0 ~Phys. Starus Solidi b 157, K13. Tanaka, K., Kohyama, M. (l998), in: 14th Int. Con$ on Electron Microscopy, (ICEM 14), Cancun, Vol. 11: pp. 581-582. Tanaka, K., Kohyama, M., Iwasa, M. ( 1 9 9 9 in: ~ Intergranular and Interphase Boundaries in Materials: Lejcek, P., Paidar, V. (Eds.). Zurich: Trans Tech, pp. 187-190. Tarnow, E., Dallot, P., Bristowe, P. D., Joannopoulos, J. D., Francis, G. P., Payne, M. C. (1990), Phys. Rev. B42, 3644. Tasker, P., Duffy, D. M. (1983), Phil. Mag. A47, L45. Taylor, W. E., Odell, N. Y., Fan, H. Y. (1952), Phys Rev. 88, 867. Teichler, H. (1989), in: Polycrystalline Semiconducrors: Grain Boundaries and Interfaces: Moller, H. J., Strunk, H. P., Werner, J. H. (Eds.). Berlin: Springer, pp. 25-33. Tersoff, J. (1988), Phys. Rev. B37, 6991. Tersoff, J . (1989), Phys. Rev. B39, 5586. Tewary, V., Fuller, E., Jr., Thomson, R. M. (1989), J . Mater. Res. 4, 309. Thibault, J., Putaux, J. L., Bourret, A., Kirchner, H. 0. K. (1989). J. Phys. 5, 2525. Thibault, J., Putaux, J. L., Jacques, A., George, A., El Kajbaji, M. ( 19901, Microsc., Microanal., Microstruci. 1, 395. Thibault, J . , Putaux, J. L., Michaud, H.-M., Baillin, X., Jacques, A,, George, A. (1991), Inst. Phys. Con$ 117, 105. Thibault, J., Putaux, J. L., Jacques, A,, George, A., Michaud,H.-M., Baillin, X. (1993),Mater.Sci. Eng. A164, 93. Thibault, J., Baillin, X., Pelissier, J., Michaud, H.-M., Putaux, J. L. (1994a), in: Defect-Inferface Interaction, Vol. 3 19: Pittsburgh, PA: MRS, pp. 215-226. Thibault, J., Rouviere, J. L., Putaux, J. L. (1994b), in: Electron Microscopy of Boundaries and Interfaces in Materials Science: Heydenreich, J., Neumann, W., (Eds.). Elbe Druckerei Wittenberg, p. 139. Thibault-Desseaux, J., Putaux, J. L. (1989), Inst. Phys. Con$ 104, 1. Thibault-Desseaux, J., Putaux, J. L., Kirchner, H. 0. K. (1989). in: Point and Extended Defects in Semiconductors: Benedek, G., Cavallini, A,, Schroter, W. (Eds.). New York: Plenum, pp. 153-164. Thomson, R. E., Chadi, D. J. (1984), Phys. Rev. B29, 889. Torrent, M. (1996), Ph.D. Thesis, University Paris VI, repon CEA-R-5733. Turan, S., Knowles, K. M. ( 1 9 9 9 ~in: Intergranular and Interphase Boundaries in Materials: Lejcek, P., Paidar, V. (Eds.). Zurich: Trans. Tech, pp. 3 13- 3 16.

7.8 References Valiev, R., Gertsman, V., Kaibyshev, O., Khannov, S. (1983), Phys. Status Solidi ( a ) 77, 97. Vanderbilt, D., Taole, S. H., Narasimhan, S. (1989), Phys. Rev. B40, 5657. Vaudin, M., Cunningham, B., Ast, D. (1983), Scr. Met. 17, 191. Vaudin, M., Handwerker, C., Blendell, J. (1988), J. Phys. C5-49, 687. Vlachavas, D., Pond, R. C. (1981), Insr. Phys. Con$ 60, 159. Vogel, E L., Pfann, W., Corey, H., Thomas, E. (1953), Phys. Rev. 90, 489. Watanabe, T. (1988), J. Phys. C5-49, 507. Weber, W. (1977), Phys. Rev. B15, 4789. Werner, J. (1985), in: Polycrystalline Semiconductors: Harbeke, G. (Ed.). Berlin: Springer, pp. 76-87. Werner, J. (1989), Inst. Phys. Con$ 104, 63. Werner, J., Guttler, H. H. (1991), J. Appl. Phys. 69, 1522. Werner, J., Peisl, M. (1985), Phys. Rev. B31, 6881. Werner, J., Strunk, H. (1982), J. Phys. CI-43, 89. Westwood, A. D., Notis, M. R. (1995), J. Matel: Res. io,2573. Wilder, J., Teichler, H. (1997), Phil. Mag. Lett. 76,83. Wolf, D. (1984a), Acta Metall. 32, 735. Wolf, D. (1984 b), Phil. Mag. A49, 823. Zhang, Y., Ichinose, H., Ishida, Y., Nakanose, M. (1993, in: Proc. 2nd NIRIMInt. Symp. on Advanced Materials (ISAM 995): Bando, Y., Karno, M., Haneda, H., Aizama, T. (Eds.). Nat. Inst. for Research in Inorganic Materials, pp. 271-273. Zook, J. D. (1980), Appl. Phys. Lett. 37, 223.

45 1

General Reading Benedek, G., Cavallini, A., Schroter, W. (Eds.), (1989), Point and Extended Defects in Semiconductors. NATO AS1 Series B-202. Grovernor, C. R. M. (1985), J. Phys. C. Solid State Phys. 18, 4079. Harbeke, C. (Ed.) (1983, Polycrystalline Semiconductors: Physical Properties and Applications. Solid State Sci. 57. Berlin: Springer. Leamy, H. J., Pike, G. E., Seager, C. H. (Eds.) (1982), Grain Boundaries in Semiconductors, Vol. 5 . Pittsburgh, PA: MRS. Moller,H. J., Strunk, H. P., Werner, I. H. (Eds.) (1989), Polycrystalline Semiconductors: Grain Boundaries and Interfaces. Berlin: Springer. Priester, L., Thibault, J., Pontikis, V. (1998), Solid State Phenomenum 59-60, 1. Proc. 1979 ASM Materials Science Seminar: Grain Boundary Structure and Kinetics (l980), Metals Park, OH: ASM Int. Proc. JIM Int. Symp. on Structure and Properties of Internal Interfaces, J. Phys., C4-46 (1985). Proc. Int. Con$ on Grain Boundary Structure and Related Phenomena, Trans. Jpn. Inst. Met. 27 (1986). Proc. Int. Congress on Intergranular and Interphase Boundaries in Materials 89, J. Phys., C1-5 1 (1990). Proc. Int. Congress on Intergranular and Interphase Boundaries in Materials 92, Mate,: Sci. Forum (1993), Vol. 126-128: Switzerland: Trans Tech. Proc. Int. Congress on Intergranular and Interphase Boundaries in Materials 94, Mater. Sci. Forum (1996), Vol. 207-209: Switzerland: Trans Tech. Proc. Int. Congress on Intergranular and Interphase Boundaries in Materials 98, MateK Sci. Forum, Vol. 294-296: Switzerland: Trans Tech. Raj, R., Sass, S. (Eds.) (1988), J. Phys., C5-49. Sutton, A. P., Balluffi, R. W. (1995), Interfaces in Crystalline Materials. Oxford: Clarendon. Yoo, M. H., Clark, W. A. T., Brian, C. L. (Eds.) (1988), Interfacial Structure, Properties and Design. Pittsburgh, PA: MRS, p. 122.

8 Interfaces

.

R Hull Department of Materials Science and Engineering. University of Virginia. USA

.

. .

.

A Ourmazd. W D Rau. and P Schwander

Institute for Semiconductor Physics. Frankfurt (Oder). Germany

. .

..

M L Green and R T Tung Bell Laboratories. Lucent Technologies. Murray Hill. NJ. USA

8.1 8.2 8.3 8.3.1 8.3.2 8.3.2.1 8.3.2.2 a.3.2.3 8.3.3 8.3.3.1 8.3.3.2 8.3.4 8.4 8.4.1 8.4.1.1 8.4.1.2 8.4.1.3 8.4.1.4 8.4.1.5 8.4.1.6 8.4.2 8.4.2.1 8.4.2.2 8.4.3 8.4.3.1 8.4.3.2 8.4.3.3 8.4.3.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 455 Interfaces between Lattice-Matched, Isostructural Systems . . . . . . 457 457 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 459 Microscopic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesoscopic and Macroscopic Structure . . . . . . . . . . . . . . . . . . . 472 Interfaces Defined by Inhomogeneous Doping . . . . . . . . . . . . . . . 474 479 Relaxation of Chemical Interfaces . . . . . . . . . . . . . . . . . . . . . Interdiffusion due to Thermal Annealing . . . . . . . . . . . . . . . . . . 480 481 Intermixing due to Ion Implantation . . . . . . . . . . . . . . . . . . . . 482 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interfaces Between Lattice-Mismatched, Isostructural Systems . . . . 483 Lattice Mismatch Strain and Relaxation Mechanisms . . . . . . . . . . . 483 Origin and Magnitude of Lattice Mismatch Strain . . . . . . . . . . . . . 483 Strain Accommodation and Relief Mechanisms . . . . . . . . . . . . . . 484 Epitaxial Layer Roughening . . . . . . . . . . . . . . . . . . . . . . . . 485 487 Interdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 Competition Between Different Relaxation Modes . . . . . . . . . . . . . 489 The Critical Thickness for Misfit Dislocation Introduction: Excess Stress . 490 Basic Concepts: Single Interface Systems . . . . . . . . . . . . . . . . . 490 491 Extension to Multilayer Systems . . . . . . . . . . . . . . . . . . . . . . Misfit Dislocation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 493 493 Kinetic Relaxation Models . . . . . . . . . . . . . . . . . . . . . . . . . 494 Nucleation of Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . . Propagation of Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . 497 Interactions of Misfit Dislocations . . . . . . . . . . . . . . . . . . . . . 499

454

8.4.4 8.4.5 8.4.6 8.5 8.5.1 8.5.2 8.5.2.1 8.5.2.2 8.5.2.3 8.5.2.4 8.5.3 8.5.4 8.5.5 8.5.5.1 8.5.5.2 8.5.5.3 8.5.6 8.6 8.6.1 8.6.1.1 8.6.2 8.6.3 8.7 8.8

8 Interfaces

Techniques for Reducing Interfacial and Threading Dislocation Densities . 500 Electrical Properties of Misfit Dislocations . . . . . . . . . . . . . . . . . 503 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Interfaces Between Crystalline Systems Differing in Composition and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Fabrication of Epitaxial Silicide-Si Interfaces . . . . . . . . . . . . . . . 507 507 Monolayers Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlayer Mediated Epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . 513 Growth of Silicon on Silicides . . . . . . . . . . . . . . . . . . . . . . . 515 Conglomeration of Silicide Precipitates . . . . . . . . . . . . . . . . . . 516 Epitaxial Elemental Metals . . . . . . . . . . . . . . . . . . . . . . . . . 518 Epitaxial Metallic Compounds on III-V Semiconductors . . . . . . . . . 519 Structure, Energetics, and Electronic Properties of M-S Interfaces . . . . 520 Epitaxial Silicide-Silicon Interfaces . . . . . . . . . . . . . . . . . . . . 520 Epitaxial Elemental Metals . . . . . . . . . . . . . . . . . . . . . . . . . 523 Intermetallic Compounds on III-V Semiconductors . . . . . . . . . . . . 524 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Interfaces Between Crystalline and Amorphous Materials: Dielectrics on Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 The Si/SiO, System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 The Si/SiO,N ,. System . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 531 Alternative Gate Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

8.2 Experimental Techniques

8.1 Introduction Any finite system is delimited by interfaces. In this trivial sense interfaces are ubiquitous. However, modern epitaxial techniques seek to modify the properties of materials by stacking dissimilar layers. “Band gap engineering”, the attempt to tailor the electronic properties of semiconductors by interleaving (many) dissimilar layers is an example of this approach. Many modern materials and devices thus derive their characteristics from the presence of interfaces, sometimes separated by only a few atomic distances. Interfaces between solids can be classified into four general categories. 1) Interfaces between lattice matched, isostructural, crystalline systems, differing only in composition (chemical interfaces). The GaAs/ A1,Gal,As system, with its very small lattice mismatch is the most technologically developed example. 2) Interfaces between isostructural, crystalline systems that differ in composition and lattice parameter. Ge,Si,, and GaAs/In,Gal,As are representative examples. 3) Interfaces between systems differing in composition and structure. Metal-silicide/semiconductor systems are important representatives of this most general class. 4) Interfaces between crystalline and non-crystalline layers, such as the Si/SiO, system. The exploration of the wide variety of possible interfaces and their properties is relatively new. Most extensively studied are the structural properties of interfaces, and much of our discussion will focus on this aspect. The electronic properties of interfaces have also been the subject of extensive research. A full treatment of these requires a separate review. In this chapter, the electronic properties will be considered primarily only in so far as the relationship to the structure is concerned. The reader is invit-

455

ed to consult other sources for a more extensive treatment (e.g. Capasso and Margaritondo, 1987). Since the literature concerned with interfaces is extensive and rapidly growing, this chapter does not aim to be an exhaustive review, even of the structural aspects of interfaces. Rather, the overall purpose is to familiarize the reader with some of the key concepts in this dynamic field.

8.2 Experimental Techniques To probe an interface, information must generally be extracted from a few monolayers of a sample buried beneath substantial thicknesses of material. This represents a severe experimental challenge. X-ray diffraction and scattering, and transmission electron microscopy are direct structural probes of buried interfaces. Xrays interact relatively weakly with matter, and are thus capable of deep penetration. For the same reason multiple scattering is generally absent, resulting in ease of interpretation. However, the interaction of Xrays with a single interface that often extends over only one or two atomic planes is also very weak. X-ray diffraction methods, pioneered by Cook and Hilliard (1 969), often thus rely on the presence of a periodic multilayer stack to produce sufficiently strong diffraction peaks (satellites), whose intensities can be related to the layer period and the structure of the interfaces present. In this way X-ray diffraction yields highly accurate data about the interfacial configuration, averaged over many interfaces. When the interface itself has a different inplane periodicity, for example when a periodic array of interfacial dislocations is present, X-ray scattering techniques can be used in conjunction with very bright synchrotron sources to investigate single inter-

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8 Interfaces

faces, with the information emanating from a large area of the interface. X-rays can also be used to make accurate lattice parameter measurements, and thus explore the accommodation and relief of strain in mismatched systems. Energetic electrons, on the other hand, interact strongly with matter, but can nevertheless propagate substantial distances, and thus emerge from samples of reasonable thickness ( = 0.5 pm). This combination makes energetic electrons highly efficient probes of buried interfaces (e.g. Suzuki, Okamoto, 1985; Kakibayashi, Nagata, 1986: Ourmazdetal., 1987a; Ichinoseetal., 1987; Tanaka, Miharna, 1988: Ou et al., 1989).The price for this, however, is increased complexity in interpretation of the data, because multiple scattering effects cannot be ignored. The transmission electron microscope (TEM) has become a standard tool in the investigation of interfaces (Ourmazd et al. 1987a; Kakibayashi, Nagata, 1986). In the case of lattice mismatched systems, the TEM reveals the presence of extended defects, and in the lattice imaging mode can yield information on the atomistic details of strain relaxation. For chemical interfaces, however, the sample structure is of little interest, and chemical information is needed to determine the interface configuration. Recently developed quantitative high resolution TEM techniques, specifically Chemical Mapping and QUANTITEM, yield quantitative chemical maps of such interfaces at near-atomic resolution and sensitivity (Ourmazd et al., 1989a, b, 1990; Ourmazd. 1993; Schwander et al., 1993, 1998: Kisielowski et al., 1995). For a general review of these methods see e.g. Baumann et al. (1995). There is a large variety of techniques that probe the optical or electronic properties of interfaces, and thus indirectly their structure

(e.g. Weisbuch et al., 1981; Tanaka et al., 1986; Tu et al., 1987; Bimberg et a]., 1987; Sakaki et al., 1987; Okumura et al., 1987). Most widely used are luminescence techniques (e.g. Weisbuch et al., 1981; Tanaka et al., 1986; Tu et al., 1987; Bimberg et al., 1987). Due to their inherent simplicity and convenience, they have been extensively applied, and in many instances the results used to optimize growth procedures. More recently, Raman scattering and photoemission spectroscopy have also been used. These techniques are most valuable when the optical or electronic properties of a layer are to be determined. However, the interpretation of such data in terms of the structure is also possible, although relatively difficult. The development of Near-field Scanning Optical Microscopy (NSOM) has led to the microscopic investigation of the optical properties of interfaces in systems that luminesce efficiently (Hess et al., 1994). Recently, electron holography, an interferometic TEM technique, has been used to map electrostatic potential distributions across interfaces between differently doped regions in silicon. In this way, electrical information about interfaces between differently doped materials - pn junctions - has been obtained with nm spatial resolution and 0.1 V potential sensitivity level (Rau et al., 1998). Ion scattering techniques, such as Rutherford Backscattering and Medium Energy Ion Scattering have also provided invaluable information on interface structure. The variety of techniques that have been applied to the study of interfaces precludes a treatment of each individual approach. We will thus describe each method to the extent needed for an adequate discussion of the topic under consideration.

8.3 Interfaces Between Lattice-Matched, lsostructural Systems

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8.3 Interfaces Between LatticeMatched, Isostructural Systems 8.3.1 Definition The (chemical) interface between two la tice-matched, isostructural materials can t uniquely defined on all length scales, prc vided each atom type occupies an ordere set of lattice sites. As an example, considr the GaAs/AlAs system. The interface is sin ply the plane across which the occupants of the Group I11 sublattice change from Ga to Al. The interfacial plane thus defined can in principle have a complex waveform, with undulations ranging from atomic to macroscopic length scales. It is thus convenient to describe an interface in terms of its Fourier spectrum, by specifying the amplitude of the undulations as a function of their spatial frequency (Warwick et al., 1990). Fig. 8-1 shows the Fourier spectrum of a “white noise” interface, with a constant roughness amplitude over all possible length scales. When a given experimental technique is used to investigate an interface, it provides information about the interfacial configuration within a certain frequency window, delimited on the high frequency side by the spatial resolution of the technique, and on the low frequency side by its field of view or spatial coherence. For any experimental technique, this frequency window spans only a small portion of the spatial frequencies needed for a realistic description of the interface. It is thus necessary to collate the information obtained from a large variety of techniques to obtain a complete picture of the interfacial configuration. This is a major challenge, because information from the atomic to the centimeter range, i.e. over eight orders of magnitude is required to provide such a description. However, when only specific properties, such as the optical or electronic properties of an interface are of

Figures-1. a) Roughness spectrum of a “white noise” interface. This spectrum specifies the amplitude of the roughness vs. the wavelength. The shortest possible wavelength is the atomic spacing. Any experimental technique samples only a limited part of this roughness spectrum. This “window” is bound by the field of view and the spatial resolution of the technique. b) Schematic representation of an interface.

concern, knowledge of a limited range of frequencies is adequate. In the case of luminescence due to excitonic recombination, for example, roughness over the exciton diameter is of primary importance, while for charge transport applications, roughness at the Fermi wavelength is of concern. The simple definition of an interface in terms of the location of the chemical constituents becomes inadequate when one or both of the parent materials are not chemi-

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8 Interfaces

cally ordered, i.e., when some of the atom types are distributed randomly on a set of sites (Warwick et al., 1990; Thomsen, Madhukar, 1987; Ogale et al., 1987). In the GaAs/A1,Ga,-,yAs system, for example, the second material is a random alloy. Thus, the Ga and A1 atoms are distributed randomly on the Group 111 sublattice, subject to the constraint that the composition, averaged over a sufficiently large region of the AI,Ga,,As should correspond to the value given by x . In such random alloys, the composition measured in two different regions of the same size will in general not be the same, with the difference being due to statistical departures from the average composition due to the finite sampling volume. Such variations i n the local composition grow dramatically as the sampling volume approaches atomic dimensions. In the limit of the sampling volume containing only one Group I11 sublattice site, the measured com-

position will of course be 0 or 1 , irrespective of the global average composition x. For this reason, the interface between two materials, at least one of which is a random alloy cannot be satisfactorily defined on all length scales. This is demonstrated in Fig. 8-2 where random alloy Alo,3G%,7Ashas been “deposited” on an atomically flat GaAs surface, and the resulting structure viewed in cross-section. Each panel uses shades of gray to show the composition averaged over a given number of atoms perpendicular to the plane of the paper. Consider Fig. 8-2, where only one atomic plane is used, Le., no averaging has been carried out. A line drawn to contain all the Ga atoms, Le. to define an interface on an atomic scale would deviate dramatically from the original GaAs surface upon which the random alloy Al,,,G%,,As was deposited. This illustrates that in the GaAs/AI,Ga,-,As system no interface can be defined on an atom by atom basis. (At-

Figure 8-2. Schematic representation of interface formed by depositing random alloy Alo,aGa, ,As on an atomically smooth GaAs surface (cross-sectional view). Only Group 111 atoms are shown. White represents pure aluminum, black pure gallium, other shades of gray intermediate compositions. In each case, the composition of the indiL idual atom columns (represented by squares) has been averaged over the “thickness” shown. When the sample is only one monolayer thick, i.e., there has been no averaging, no continuous line can be drawn to contain only gallium (or aluminum) atoms, illustrating that an interface cannot be defined on an atom-by-atom basis. Only as the “thickness” increases, does the interface become well defined.

8 . 3 Interfaces Between Lattice-Matched, lsostructural Systems

tempts to image interfaces by tunnelling microscopy must be viewed in this light.) Only as the “thickness” over which the composition is averaged increases, does an isocomposition line approach the initial GaAs surface. For the Al0,,Gao,,As system, the isocomposition line becomes essentially indistinguishable from the original GaAs surface when the composition is averaged over = 30 atoms per column. These considerations apply generally, regardless of whether the interface is viewed in “cross-section” as in Fig. 8-2, or in “plan-view”.

8.3.2 Structure In this section we attempt to outline how a variety of techniques may be used to gain information about the configuration of a chemical interface over a wide range of spatial frequencies. The discussion is centered on the GaAs/AlGaAs system, because it is technologically advanced and has been extensively investigated. Although the microscopic structure of this interface can now be determined quantitatively, our knowledge of its structure over other length scales remains qualitative. Nevertheless, the discussion illustrates the challenge of describing an interface over a wide frequency range, and the importance of a critical appreciation of the way different techniques provide insight into the properties of an interface.

8.3.2.1 Microscopic Structure In the absence of catastrophic crystal growth, the structure of an interface between two lattice-matched, isostructural crystals is uninteresting. For example, the atoms continue to occupy zinc-blended sites as a perfect semiconductor heterointerface is approached and crossed. In seeking to determine the atomic configuration at such an interface, one is in reality asking a chem-

459

ical rather than a structural question; one is attempting to learn which atom sits where, rather than where the atoms sit. X-ray diffraction techniques have been applied to samples containing periodic stacks of chemical interfaces. Careful fitting of the satellite peak intensities due to the periodic compositional modulation elucidates the overall features of the interface configuration. Common to many experiments is the finding that the interfacial region includes a few monolayers whose composition is intermediate between the neighboring materials (Fleming et al., 1980; Vandenberg et al., 1987). In the case of systems such as Ino,,8Gao,52As/InP, which are latticematched only at one composition, the presence of a region of intermediate composition also entails the introduction of strain. In an elegant series of experiments, Vandenberg and Panish have shown how the details of the growth procedure can modify the nature of the interfacial layer and the concomitant strain (Vandenberg et al., 1988, 1990). The TEM in its lattice imaging mode can in principle reveal the local atomic configuration of an interface. Conventional lattice imaging, however, produces a map of the sample structure, and as such is not a useful probe of chemical interfaces. Below, we briefly describe the way the TEM may be used to obtain chemicaI information at near atomic resolution and sensitivity. The combination of “chemical lattice imaging” and digital pattern recognition quantifies the information content, and hence the composition of individual cells of material = 2.8 x 2.8 x 75 A in volume. Chemical Lattice Imaging

In the modem High Resolution TEM (HRTEM), a parallel beam of energetic electrons is transmitted through a thin sample to

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8 interfaces

produce a diffraction pattern (Spence, 1988).The phases and amplitudes of the diffracted beams contain all the available information. Part of this information is passed through an aperture and focussed by the objective lens, causing the beams to interfere and produce a lattice image. In general, most of the reflections used to form a lattice image come about because of the lattice periodicity, and are relatively insensitive to the exact occupancy of the lattice sites. We name such reflections structural. However, certain reflections, such as the (200) in the zinc-blende system, are due to chemical differences between the occupants of the different lattice sites, and contain significant chemical information (Ourmazd et al., 1986, 1987a; Ourmazd, 1989. 1993). Such chemical reflections are i n general weaker than the “strongly allowed” structural reflections, and the latter usually dominate the information content of lattice images. However, two factors, multiple scattering and lens aberrations, usually considered disadvantages of HRTEM, allow one to select and enhance the relative contribution of the weaker chemical reflections to lattice images. When the electron beam enters a crystal along a low symmetry direction, a number of reflections are excited, exchanging energy among themselves as they propagate through the sample. To first order, this multiple scattering process may be viewed as the scattering of electrons from the undiffracted beam to each reflection, and their subsequent return. Structural reflections are strongly coupled to the undiffracted beam, and thus exchange energy with it rapidly as they propagate through the sample. This energy exchange lower for the more weakly coupled chemical reflections. Because of this “pendellosung” effect, at certain sample thicknesses a chemical reflection can actually have a larger amplitude than its structural counterpart (Fig. 8-3). Appropri-

I -.-

200

0

c C

L

C

0.1

100 Thickness ( A )

200

Figure 8-3. Variation of beam amplitudes with thickness (pendellosung) for InP and InGaAs. The vertical lines show a suitable thickness window for maximum periodicity change, and hence chemical sensitivity across an InPhnGaAs interface.

ate choice of sample thickness can thus enhance the chemical information content of the lattice image. Moreover, the severe aberrations of electromagnetic lenses impart the character of a bandpass filter to the objective lens, whose characteristics can be controlled by the lens defocus (Spence 1988: Ourmazd et al. 1986, 1990). Thus, judicious choice of defocus allows the lens to select, and thus further enhance the contribution of the chemical reflections to the image. To obtain chemical lattice images of compound semiconductor heterointerfaces in practice, advantage is taken of the chemical sensitivity of the (200)reflections (Fig. 8-3).


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The sample is viewed in the (100) orientation, and the (200) (chemical) and (220) (structural) reflections are used to form an image. The sampel thickness and lens defocus are chosen to maximize the change in the frequency content of the lattice image across the interface. Thus the chemical information in the sample is encoded into periodicity information in the lattice image with maximum sensitivity (Ourmazd et al. 1987a, 1990).

Atomic Configuration of Chemical Interfaces Figure 8-4a is a structural lattice image of an InP/InGaAs interface. Such (structural) images are widely used to investigate the nature of chemical interfaces. It is of course true that even these structural images reveal, to some extent, the chemical change across the interface through the change in the background intensity. The question is whether this sensitivity is sufficient for these images to reveal the atomic details of the interfacial configuration. Fig. 8-4b is the same as Fig. 8-4a, except that the line marking the position of the interface is removed. The interface position and configuration are now less clear. This emphasizes the limited chemical sensitivity of structural images. Fig. 8-4c is a chemical lattice image of the same atom columns, obtained under optimum conditions for chemical sensitivity (Ourmazd et al., 1987a). The InP is represented by the strong (200) periodicity (2.9 A spacing), while the InGaAs region the (220) periodicity (2 A spacing) is dominant. Clearly, the interface is not atomically smooth, the roughness being manifested as the interpenetration of the (200) and (220) fringes. While this demonstration established the inadequacy of normal structural lattice images to reveal the interfacial configuration,

Figure 8-4. a) (110) (structural) image of an InPIInGaAs interface. The line draws attention to the interface. b) Same image without line. c) Same area of interface imaged along (100) under chemically sensitive conditions. Note the interpenetration of InP (200) and InGaAs (220) fringes, indicating interfacial roughness.

it does not necessarily imply that all semiconductor heterointerfaces are rough. In the case of the technologically more mature GaAs/AlGaAs system, the photoluminescence linewidth, and particularly the socalled monolayer splitting of the photoluminescence lines have been interpreted in terms of atomically abrupt and smooth interfaces,with the spacing between interfacial steps estimated at several microns (see below). It is thus important to examine microscopically GaAs/AIGaAs interfaces shown by luminescence to be of the highest quality.

462

8 Interfaces

Figure 8-5 is a chemical lattice image of a high quality GaAs/A1,,,,G%,63As interface, grown with a two minute interruption at each interface (Tu et al., 1987). The sample thickness and imaging conditions correspond to maximum chemical sensitivity, reflected in the strong change from (220) to (200) periodicity across the GaAs/AlGaAs interface. Visual examination of the image directly reveals the presence of interfacial roughness. Thus, even at this qualitative level of inspection, interfaces of the highest optical quality appear microscopically rough. However, the general practice of evaluating lattice images by visual inspection is subjective and unsatisfactory. We describe below a digital pattern recognition approach, which quantifies the local information content of lattice images, leading to their quantitative evaluation (Ourmazd et al., 1989a, b, 1990: Ourmazd, 1993).

Figure 8-5. Chemical lattice image of GaAs/Al, 37 Gk,,,As quantum well produced after two minutes of growth interruption at each interface. Careful inspection reveals interfacial roughness.

Quantification of Local Information Content of Images The information content of a lattice image is contained in its spatial frequency spectrum, or alternatively, in the set of patterns that combine in a mosaic to form the image (Ourmazd et al., 1989a, b). In practice, the information content is degraded by the presence of noise. The quantitative analysis of the information content thus requires three steps: (i) the assessment of the amount and the effect of noise present; (ii) the identification of statistically significant features; and (iii) quantitative comparison with a template. A primary virtue of an image is that it yields spatially resolved information. Thus, whether these tasks are carried out in Fourier- or real-space, the retention of spatial resolution, that is, the local analysis of the information content is of paramount importance. In a chemical lattice image, the local composition of the sample is reflected in the local frequency content of the image, or alternatively, in the local patterns that make uptheimage(Ourmazdetal., 1987a, 1989a, b). Thus local analysis of the image is equivalent to local chemical analysis of the sample. Local analysis of an image can be most conveniently affected by real-space rather than Fourier analysis. Real-space analysis proceeds with the examination of the information content of a unit cell of the image. When the integrated intensity is used to characterize a cell, information regarding the intensity distribution within the cell is not exploited. In one dimension this is analogous to attempting to identify a curve from the area under it, which would yield an infinite number of possibilities. Here we describe a simple procedure that exploits the available information more fully. The task is carried out in several steps. First, perfect models, or templates are


adopted from simulation, or developed from the data, which serve to identify the ideal image of each unit cell type. When the template is extracted from experimental images, it is obtained by averaging over many unit cells to eliminate the effect of noise. For example, several unit cells of Fig. 8-5 not lying at the interface are averaged to produce the templates for GaAs and A10.37Gao,63As shown in Fig. 8-6a. Second, an image unit cell of a particular size is adopted, and divided into an n x n array of pixels, at each of which the intensity is measured. Typically n = 30, and thus 900 intensity measurements are made within each unit cell. Third, each unit cell is represented by a multidimensional vector, whose components are the n (usually 900) intensity values obtained from the cell. The ideal image unit cell for each material is now represented by a template, which in turn is represented by a vector R'. For example, the ideal image unit cells of GaAs and A10,37Gao,63As are characterized by the two respectivevectors RLaAsand Ralo,3,Gao,63As, ly (Fig. 8-6 b).

463

Next, the amount of noise present in the experimental image is deduced from the angular distributions of the real (that is, noisy) unit cell vectors RGaAs and RAlo,s,Gao,63As about their respective templates. The noise in Fig. 8-5 is such that, away from the interface, the R G a A ~ and R A 1 0 . 3 7 G ~ . 6 3 A ~ form similar normal distributions around their respective template vectors R L a A s and Ra10,37Ga0,63As. The standard deviation CJ of each distribution quantifies the noise present in the images of GaAs and A10,37Ga0,63A~ (Fig. 8-6c). A unit cell is different from a given template, with an error probability of less than 3 parts in lo3, it its vector is separated from the template vector by more than 3 0.With 3.9 x3.9 A2 image unit cells, the centers of the distributions for the GaAs and A10,37Ga0,63A~ unit cells shown in Fig. 8-6 are separated by 12 CJ, which means that each unit cell of GaAs and A10,37Gao~,3As can now be correctly identified with total confidence. A representation of the results of the vector pattern recognition analysis of Fig. 8-5 is shown in Fig. 8-7. The image is divided into 2.8x2.8 A2 cells, each of

Figure 8-6. a) Averaged, noise-free images of GaAs (left) and A1037 Gao,&S (right). The unit cells used as templates for pattern recognition are the dotted 2.8 8, squares. b) Schematic representations of the template vecthe distribution of RGaAs and R i I G a A sabout , them, and an interfacial vector R'. tors RLaAsand R~,o,,,GQ,,,As, c) Schematic representation of the distribution produced by the GaAs and A10,37Gk,63A~ unit cells about their templates. Note that the angular position of R' denotes the most likely composition only. The actual composition falls within a normal distribution about this point.

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8 Interfaces

Figure 8-7. Three-dimensional representation of the analyzed lattice image of A10,3@3,,,63AS grown on GaAs after a 2 min interruption. The unit cells are 2.8 A squares. The height of each cell represents the angular position of its vector R with respect to the template vectors, which are about 12 u apart. Yellow and blue mark those ~ respectively. Green, magenta, and red cells which fall within 3 o of the GaAs and AIo 3 7 G k , 6 3 Atemplates, represent 3 o bands centered about 3.6, and 9 u points from GaAs. Outside the yellow and blue regions, the aluminum content of each unit cell is intermediate between GaAs and AIo,,,Gao,6,As, with confidence levels given by normal statistics

which is placed at a height representing the angular position of its vector. The yellow and blue cells lie within 3 o of R,,, and RAlo3,Gaoh3As respectively, while the other colors represent 3 to 5 , 5 to 7 , and 7 to 9 o bands (Ourmazd et al., 1989a, b: Ourmazd, 1993). We have now outlined a simple approach capable of quantitatively evaluating the local information content of images made up of mosaics of unit cells. This method exploits all the available information to determine the amount of noise present, is sophisticated in discriminating between noise and signal, identifies statistically significant features, and allows quantitative comparison with templates. Below, we discuss how, in the case of chemical lattice images, the

local information content is related to the local composition of the sample. A lattice image is locally analyzed to gain information about the local atomic potential of the sample. Under general dynamical (multiple) scattering conditions, the electron wavefunction at a point on the exit face of the sample need not reflect the sample projected potential at that point. The emerging electron wave is further convoluted with the aberrations of the lens before forming the image. There is no general relation connecting the local details of a lattice image to the local atomic potential in the sample (Spence, 1988). In chemical imaging, we are concerned with the way that a compositional inhomogeneity is imaged under conditions appro-


priate for chemical sensitivity, and how the pattern recognition algorithm extracts information from a chemical lattice image. For sample of reasonable thickness (e300 A at 400 kV), as the AI content of homogeneous Al,Ga,,As is changed from 0 to 0.37, the vector R,&,Ga,_,As rotates linearly from RbaAs to R L I ~ . ~ ~ G(Ourmazd ~ ~ . ~ ~ A et ~al.9 1990). Thus, in homogeneous material, the composition of a unit cell can be directly deduced from the angular position of its vector R with respect to the templates. In general, R deviates from the plane containing the template vectors, and the projection of R on this plane yields the composition. The confidence levels associated with such measurements depend on the amount of noise present, and can be deduced from normal statistics. In an inhomogeneous sample, this simple procedure requires justification. The problem can be formulated as follows. Given a “chemical impulse” of a specific shape, such as a column of AI atoms imbedded in GaAs (a &function), an abrupt interface (a &function), or a diffuse interface (say an error function), what is the shape of the impulse on the analyzed chemical image? Or, alternatively, what region of the sample contributes to the information content of an im-

465

age unit cell? By reciprocity, these two formulations are equivalent. This problem is essentially similar to determining the response function of a system. The effect of the response function can be determined by analyzing images of samples containing various impulses, simulated under conditions appropriate for chemical imaging (Ourmazd et al., 1989a, b, 1990). The appropriate conditions are chosen from a bank of simulated images that contain the particular impulse under consideration. For example, the simulated images of an abrupt GaAs/A1,,,,GaO,,,As interface (&function) show, that in this case, the appropriate conditions correspond to sample thickness and lens defocus values of = 170 and =- 250 A, respectively. Such analysis shows that under appropriate chemically sensitive conditions nonlocal effects due to dynamical scattering and lens aberrations are negligible (Ourmazd et al., 1990). This is illustrated in Fig. 8-8, where at the chemical image of a column of AI imbedded in GaAs (a &function) is simulated and then analyzed; the input impulse and the analyzed response identical. The response function is essentially determined by the periodicity of the chemically sensitive reflection, which in the case of the zinc-blende structure is the (200)

Figure 8-8. a) Simulated image and b) analyzed image of a series of &functions of aluminum, embedded in GaAs. Sample thickness: 170 A, defocus: - 250 A.

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8 Interfaces

periodicity. This means that in this structure, the composition of a region 1/4 of the crystal unit cell in cross-section and = 15 atoms high can be directly determined (Baumann et al., 1992). Several other questions regarding the practicality of the approach outlined above, such as the effects of geometrical imperfections in the sample, and photographic nonlinearities have been considered elsewhere (Ourmazd et al., 1990: Ourmazd, 1993). What emerges is the conclusion that the combination of chemical lattice imaging and vector pattern recognition can lead to quantitative analysis of chemical interfaces at near-atomic resolution and sensitivity. Below, we apply these techniques to quantify the composition change across interfaces of the highest quality.

Quantitative Chemical Maps Fig. 8-7 is the analyzed chemical lattice image of the GaAs/A10,3,Gao,6,Asinterface of Fig. 8-5. The height of a unit cell represents the angular position of its vector R with respect to the template vectors, and the color changes represent statistically significant changes in composition over and above random alloy statistics. This representation allows a quantitative display of the noise and the composition at each Group I11 atomic column = 30 atoms thick. The compositional change from 0 to 0.37 corresponds to changing a column of 30 Ga atoms to one containing = 19 Ga and 1 1 A1 atoms. It turns out that the replacement of one or two Ga atoms with A1 can be detected with 60% or 90% confidence, respectively (Ourmazd et al., 1990). This demonstrates that Fig. 8-7 is essentially a spatial map of the composition, at near-atomic resolution and sensitivity. Although luminescence shows this interface to be of the highest quality (Tu et al. 1987), it is clear that its

atomic configuration is far from “ideal”. The quantitative chemical map of Fig. 8-7, which is typical, shows that the transition from GaAs to A10,3,Gao,63AS takes place over = 2 unit cells, and that the interface contains significant atomic roughness. It is important to note that the region of sample analyzed in Fig. 8-7 is =30 atoms thick, and thus random alloy roughness is expected to be at a negligible level. Also, because in our analysis the statistical fluctuations in the local composition due to random alloy statistics contribute to the “noise” in the AlGaAs region, only roughness over and above the random alloy component is evaluated as statistically significant. At the level of detail of these composition maps, the assignment of values for interfacial imperfections, such as transition width, roughness, and island size, is a matter of definition. Also, without extensive sampling, caution is required in deducing quantitative values for the spacing between interfacial steps, however they are defined. Nevertheless, it is clear that significant atomic roughness at the = 50 A lateral scale is present.

The QUANTITEM Technique In the absence of chemical reflections, it is not obvious how to distinguish changes in composition from changes in thickness; each affects the local lattice image pattern. A typical example is the Si/GeSi system, where the germanium atoms do not occupy an ordered set of sublattice sites, but form a random alloy. Figure 8-9 shows a simulated lattice image of a Geo,25Sio.,5quantum well in a wedge-shaped sample, where similar image patterns are obtained both in the quantum well and in pure silicon, but at different thicknesses. In order to determine quantitative chemical information from such lattice images, a more general approach than chemical lattice imaging is


467

Figure 8-9. Simulated lattice image of a Geo,25Sio,,5quantum well in a wedgeshaped sample. The thickness gradient runs at 35’ to the vertically oriented quantum well. Note that similar image patterns are obtained both in the quantum well and in pure silicon (see circles), but at different thicknesses.

needed. In the following, we show how chemical information can be obtained at near-atomic resolution by the so-called QUANTITEM technique in systems, that display no chemical reflections. In high resolution TEM, a crystalline sample is usually imaged along a low indexed zone axis. The resulting lattice image can be interpreted as a fingerprint of the sample (Coulomb) potential projected along the zone axis. Quantitative HRTEM attempts to measure the atomic potential. This requires the determination of two quantities: the projected potential, and the corresponding uncertainty, or measurement error. Fundamentally, high resolution lattice imaging is characterized by a highly nonlinear relationship between the image intensity distribution Z(x, y ) and the sample projected potential P (x, y ) , which may generally be described by

’

I ( x , y > = F ( P ( x ,Y ) , Si>

’

(8-1)

Strictly speaking, this relies on the projected potential approximation, in which the “vertical” position of an atom within a column is considered immaterial (Spence, 1988).

where Sidenotes all the imaging parameters [defocus, accelerating voltage, etc.; see, e.g., Spence ( 1 SSS)]. F is a complicated, and in general unknown function, which relates the image intensity to the sample projected potential. In this description, changing the imaging conditions leads to a different region of the function F. For a single image, obtained under a particular set of imaging conditions

I (x, y>= F ( P (x, y ) , So)

(8-2)

or in other words

I (x, Y ) = F O P (x, Y > >

(8-3)

Real-space analysis attempts to derive the function from the information contained in the image alone, thus directly relating the image intensity to the projected potential for each image. At present, this is only possible for crystalline samples without extended defects. Here we consider the two real-space methods: chemical mapping and QUANTITEM. In the case of chemical mapping, the nonlinear relationship between the projected potential and the intensity distribu-

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8 Interfaces

tion is used to define chemically sensitive conditions. These select appropriate “operation windows” tuned to maximum sensitivity to composition changes and minimum (“zero”) sensitivity to thickness changes. QUANTITEM, on the other hand, is applicable when such operation windows of enhanced chemical sensitivity are not available or are too small to be used in practice. Maurice et al. ( 1 997) have shown that both methods may be viewed as limiting cases of a more general approach. The domain of applicability of these techniques must be determined for each materials system. This is done by simulation. Chemical mapping and QUANTITEM are tools to study the path traced out in multidimensional space by the vector tip R of the image unit cells (see Sec. 8.3.2.1) within the field of view. Alternatively, QUANTITEM attempts to determine the path 9, while chemical mapping finds conditions under which 9 is particularly simple. Chemical mapping requires the presence of chemical reflections and operation within “chemical mapping windows”. Under such circumstances, the path may be easily parameterized i n terms of composition changes alone. For systems without chemical reflections, the path can often be approximated by relatively simple curves or surfaces. Indeed, for many materials and zone axes, the path is nearly an ellipse, whose exact shape depends on the imaging conditions. This stems from the physics of dynamical scattering and image formation. It can be understood by establishing a link between real-space techniques using vector pattern recognition and conventional theory of dynamical scattering and nonlinear image formation (Kisielowski et al., 1995; Schwander et al., 1998). The current implementation of QUANTITEM determines this ellipse, i.e., the function p ,for each image. This relates

the image intensity to the sample projected potential P. QUANTITEM allows measurement of the compositional or thickness variations in the absence of chemical reflections. As an important practical point, it does not require specific imaging conditions (e.g., defocus). QUANTITEM parameterizes the path in terms of projected potential or “reduced thickness” (thickness in units of extinction distance, see Sec. 8.3.2.1). Thus the thickness variations can be mapped when the extinction distance (composition) is known, or inversely the extinction distance (composition) when the thickness is known (Schwander et al., 1993). The principle behind QUANTITEM can be understood by first considering a sample of uniform composition but changing thickness. This is usually the case in practice, since TEM samples are always wedgeshaped and atomically rough. The basic idea is to determine the function p ( P ) relating the image intensity I to the projected potential P (in this case, simply the sample thickness). This function is obtained as follows: Each unit cell represents a random sampling of the effect of the projected potential P on the image intensity I. These samples cover all values assumed by the potential over the field of view - precisely the range needed to relate I to P for the image in hand. The function, which is very nearly an ellipse, is periodic with sample thickness, with a periodicity given by the extinction distance. This stems from the periodicity of the “pendellosung” oscillations, described in Sec. 8.3.2.1. Given the random samples for the entire range of the function p ( P ) , and knowing the period of the function in terms of the extinction distance (i.e., sample thickness), the function p ( P ) can be entirely determined from a single lattice image, without knowledge of the imaging conditions.


469

Figure 8-10. Lattice image unit cells and their vector representation R' for three different sample thicknesses. The cloud of points represents the tips of vectors extracted from an experimental image of a wedge-shaped silicon sample. The path described by the image vectors represents the function F o ,which relates the image intensity to the sample projected potential.

In practice, one proceeds as follows: The path described by the tip of the unit cell vector R can be determined by plotting the vectors representing the individual image unit cells over the field of view. simply represents the curve that is described by the vector tips. This is graphically shown in Fig. 8-10, obtained from an experimental image of a wedge-shaped silicon sample. The problem of calibrating this path in terms of (known) changes in projected potential can be tackled in different ways. Here we concentrate on lattice images that are taken in high symmetry directions of semiconductor materials in the absence of chemical reflections. In such cases, a convenient parameterization of the path yields a path variable, which changes linearly with the projected potential. As an example, consider silicon in the (1 10) zone axis. The points in Fig. 8- 11 each represent the tip of an image unit cell vector obtained from a simulated image of a silicon wedge-shaped sample. For thickness changes of up to 3/4 of the period (extinction distance), the path is well-approximated by an ellipse. This strongly suggests using the ellipse phase angle @e as the path variable. Here, @e is defined by the relation X=cos @c and Y=sin @ e , where (X, Y) is a point on the ellipse plane. Figure 8-12a shows the dependence of the ellipse phase

angle @e on the sample projected potential in three different zone axes, at five different values of lens defocus. The ellipse angle @e changes linearly with sample thickness at a rate given by the extinction distance. This remarkable relation between the sample thickness and the ellipse phase angle has a simple physical explanation. When a high energy electron beam propagates through a crystal along a zone axis, only a few eigenstates (i.e., Bloch waves) are excited (Kambe et al., 1974). It turns out that an exact ellipse

SIMULATION At = 3.8 A FIT

- ELLIPSE

Si 4 10>

X = A COS @e Y = B sin @e

I

Figure 8-11. QUANTITEM analysis of a simulated image of a silicon wedge. Each point is the tip of a vector representing an image unit cell. As the sample thickness increases in 3.8 A (0.38 nm) increments, the unit cell vectors describe a path, which is almost exactly an ellipse for thickness changes of up to threequarters of an extinction distance.

470

8 Interfaces

12 0 0 '0 0 6~c111>v 4

0

Q

0 0

0

D

Q

b

0

Gexsil-x

10

8

@e

4: 6 ~=.05 0 xr.10 A X=.15 xn.20 X X=.25 X

2-

0.0

(a)

0.2

0.4

0.6

t/t

0.8

1.0

1.2

1.0

(b)

1.2

1.4

4 2 1.6

tit

Figure 8-12. Variation of the ellipse phase angle @ e with sample thickness t , normalized to the extinction distance E, simulated for a) silicon for three different zone axes and five different defocus values, and b) Ge,Si,, for different germanium concentrations s.Note the strong overlap of the points, indicating a universal relation between the variation in @ e and the projected potential in these systems, irrespective of sample thickness, projection direction, and lens defocus

results when only two Bloch waves are excited. As shown by Kisielowski et al. (1 995) and Schwander et al. (1998), this is an excellent approximation for semiconductors and low-index zone axes, particularly when strong chemical reflections are absent. Consider next a random alloy such as Ge,Si,-, . Alloying has twoeffects: First, the elliptical path described by the vector is changed. Second, the path period, i.e., the extinction distance is altered. In many cases, the first-order change concerns only the extinction distance, with the vector path essentially remaining unchanged. For small changes in concentration, the vectors representing different compositions lie, to within noise, on a single ellipse. This is demonstrated in Fig. 8-12b for Ge,Si,, over the concentration range 0 < x < 0.25. Under such circumstances, the primary change due to the concentration is reflected in the extinction distance. As a corollary, the changes in concentration and thickness move the vector along the same path, altering only the

path traversal rate. This means these composition and thickness changes must be treated on an equal footing; there are no "chemically sensitive windows". We will now describe how QUANTITEM measures composition. Obviously the effect of sample thickness must be taken into account, as it may well vary locally and over the field of view. Consider the experimental lattice image of a Si/Ge0.,,Sio,,,/Si quantum well shown in Fig. 8-13a. First, an ellipse is fitted to the data points representing the image unit cell vector tips. The ellipse phase angle is then used as the key parameter representing changes in the sample projected potential. At this stage, a change in the phase angle includes contributions both from thickness and compositional variations. In order to determine and "subtract off' the effect of thickness changes, the thickness at the target unit cell must be determined. This is done by an interpolation approach, as described next. First, the local thickness changes are mapped in regions of

471


known composition, and the sample wedge is fitted to a two-dimensional model. This model is then used to interpolate the thickness across regions of unknown composition (e.g., the interface). At a “target” unit cell of unknown composition, the effect of thickness is subtracted and the remainder associated with a change in concentration. This approach leads to chemical maps of the type shown in Fig. 8-13.c. A primary feature of QUANTITEM is its ability to measure the (crystalline) sample thickness in samples of uniform composition at the level of one unit cell and at a precision approaching a few monolayers (see Table 8-2). Below, we describe how QUANTITEM can be used to measure the atomic roughness of buried interfaces from samples

Table 8-1. Crystal structures and lattice constants.

Figure8-13. a) Experimental lattice image of a SilGeo,,,Sio,,,/Si structure. b) Map of phase angle @e over the sample. Height represents @ e , which in areas of constant composition gives the sample thicknesses. Note the significant changes in sample thickness in the silicon region on either side of the GeSi layer. Inset: Schematic representation of the way QUANTITEM interpolates the sample thickness over regions of unknown composition and separates @e into parts due to thickness and composition changes. In practice, the interpolation scheme is based on a two-dimensional, surface-fitting procedure. c) QUANTITEM composition map of the quantum well. Height represents composition. The one-sigma ( a ) error bar is shown.

Material

Crystal structure

AI Ag a-Fe CoGa NiGa CoAl NiAl LuAs ScAs ErAs Nisi, COS2 CrSi, Pd,Si ErSi,, TbSi2, YSi,, a -Fe Si,

f.c.c. f.c.c. b.c.c. CSCl CSCl CSCl CSCl NaCl NaCl NaCl CaF, CaF, C40 (H) C22 (H) C32 (H) C32 (H) C32 (H) T

4.05 4.09 5.74 2.88 2.89 2.86 2.89 5.68 5.46 5.74 5.41 5.31 4.428 6.49 3.798 3.841 3.842 2.684

A l l (0)

9.863

diamond zincblende zinc blende

5.43 5.65 5.89

P-FeSi, Si GaAs InP

Lattice constant

-

-

-

6.363 3.43 4.088 4.146 4.144 5.128 (b) 1.791 (c) 1.833 -

472

8 Interfaces

Table 8-2. Schottky barrier heights of epitaxial silicides. Silicide

Orientation

Substrate

Interface structure

SBH (eV) n-type p-type

Nisiz Nisi, Nisi, Nisi, Nisi, CoSi, CoSi, CoSi, CoSi, a

Type A S i ( ] 1 I ) 7-fold Type B S i ( l 1 1 ) 7-fold (100) Si(100) 6-folda ( 100) Si ( 100) 7-fold { 11 1 } A ( 110) Si ( 1 IO) 7-fold { 1 1 1 ) A Type B Si( 1 1 1 ) 8-folda Type B Si(l11) “X3lb (100) Si(100) 8-folda (110) S i ( l l 0 ) &folda

0.65 0.79 0.40 0.65 0.65 0.67 0.27 0.71 0.70

0.47 0.33 0.73 0.45 0.45 0.44

0.71 0.41 0.42

Tentative; Sullivan et al. (1993).

in plan view, revealing topographic maps of interfacial roughness in Si/SiO, . Due to its key role in integrated circuit technology, the Si/SiO, system is one of the most studied interfaces (see, e.g., Helms and Deal, 1988). Microscopic roughness at this interface [e.g., MOSFET gate oxides with thicknesses below 100 A (10 nm)] affects the carrier mobility and device reliability. Valuable information on the atomic configuration of this interface has been obtained by lattice imaging in cross-section (see, e.g., Goodnick et al., 1985).However, cross-sectional investigation provides only a projected view of the interface. This is a limitation encountered whenever information from buried interfaces is required. QUANTITEM measures the thickness of the crystalline part of the sample with any amorphous overlayer adding noise. If the crystalline part is bounded by two identical interfaces (such as in a SiO,/Si/SiO, sandwich), or if one of its interfaces has known or negligible roughness, QUANTITEM may be used to measure the atomic configuration of the interface of interest (Fig. 8-14). Figure 8-14b is a plan view lattice image, with Figs. 8-14c and 8-14d repre-

senting QUANTITEM maps of the S O 2 / Si( 1OO)/SiO, sandwich formed by chemical etching of the silicon sample from both sides and subsequent formation of a native oxide. In the three-dimensional representations, height represents the local roughness of the two interfaces (top and bottom), viewed in superposition. Such roughness maps can be used to calculate the autocorrelation function which quantifies the spatial extent of interfacial undulations. In summary, QUANTITEM yields composition and/or thickness maps of crystalline samples devoid of chemical reflections. Such maps reveal microscopic interfacial roughness over spatial distances extending from atomic dimensions to a few hundred nanometers.

8.3.2.2 Mesoscopic and Macroscopic Structure Due to the limited field of view of direct microscopic techniques, they cannot be used to establish the interfacial configuration over mesoscopic (micrometer) or macroscopic (millimeter) length scales. To make further progress, it is necessary to use indirect methods to gain insight into the interfacial configuration. Such techniques attempt to determine the interfacial structure through its influence on other properties of the system, such as its optical or electronic characteristics. Fundamental to this approach is the premise that it is known how the structure affects the particular property being investigated. In practice, this is rarely the case. “Indirect” experiments thus face the challenge of simultaneously determining the way a given property is affected by the structure and learning about the structure itself. Because a direct correlation is thought to exist between the structure of a thin layer and its optical properties, luminescence


473

Figure 8-14. a) Schematic representation of the way in which roughness at buried interfaces may be investigated by QUANTITEM, which measures the thickness of the crystalline part of the sample. b) Plan view lattice image of a SiO,/Si (lOO)/SiO, sandwich. c) QUANTITEM topographic map showing the roughness of the two interfaces (top and bottom) in superposition. Height represents roughness. The steep drop in the left corner is due to cleavage of the thin foil at the edge. d) Same as c) but with magnified roughness ( z ) scale.

techniques have been extensively applied to investigate the structure of semiconductor interfaces (Weisbuch et al., 1981; Tu et al., 1987; Bimberg et al., 1987; Thomsen and Madhukar, 1987). In photoluminescence (PL) the carriers optically excited across the band gap form excitons and subsequently recombine, often radiatively. The characteristics of a photon emitted due to the decay of a single free exciton reflect the structural properties of the quantum well, averaged over the region sampled by the recombining exciton. In practice, the observed signal stems from a large number of recombining excitons, some of which are bound to defects. The challenge is to extract informa-

tion about the interfacial configuration from the PL measurements, which represent complex weighted averages of the well width and interfacial roughness sampled by a large collection of excitons. The recognition that PL cannot easily discriminate between the recombination of free excitons and those bound at defects has led to the application of photoluminescence excitation spectroscopy (PLE), which is essentially equivalent to an absorption measurement, and thus relatively immune to complications due to defect luminescence. The photoluminescence spectrum of a typical single quantum well t.50 A ( 5 nm) wide, grown under standard conditions,

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474

8 Interfaces

consists of a single line -4.5 meV wide, at an energy position that reflects the well width and the barrier composition (Tu et al., 1987). This linewidth is significantly larger than that of a free exciton in high quality “bulk” GaAs (= 0.2 meV), indicating additional scattering, presumable partly due to interfacial roughness. When growth of the layer is interrupted at each interface, and the next layer deposited after a period of tens of seconds, the PL spectrum breaks into two or three sharper lines each = 1.5 meV wide. This reduction in the PL linewidth is ascribed to a smoothing of the interfaces during the growth interruption. Perhaps more strikingly, however, is the fact that the two or three lines obtained from a single quantum well were, for a long time, often assigned to excitonic recombination in different regions of the quantum well under the laser spot, within each of which the well was claimed to be an exact number of atomic layers thick. Thus the different lines were each thought to arise from recombination within “islands” over which the interfaces were atomically smooth. This model rested essentially on the premise that the several PL and PLE peak energies and their separations correspond to wells exactly an integral number of monolayers (MLs) thick. This interpretation of the luminescence data thus advocated the existence of atomically perfect (Le., atomically smooth and abrupt) interfaces. On this basis, a quantum well of nominal thickness n in fact consists of regions (islands), within each of which the thickness is exactly ( n- l ) , n, or ( n + 1 ) MLs, between which the interfacial position changes abruptly by 1 ML. These islands were sometimes claimed to be as large as 10 ym in diameter (Bimberg et al., 1987), but were generally thought to lie in the micrometer range (Miller et al., 1986; Petroff et al., 1987), and in any case to be much larger than the exciton diameter (= 15 nm).

The model of large, atomically smooth interfaces was not supported by chemical mapping, which clearly revealed significant microscopic roughness. This led to a significant controversy, whose final resolution involved a number of elegant PL and PLE experiments (Warwick et al., 1990; Gammon et al., 1995), culminating in direct PL experiments by near-field scanning optical microscopy (NSOM). Due to its superior spatial resolution, NSOM is able optically to excite and detect luminescence from very small regions of a sample. Hess et al. (1 984) were thus able to show that each PL line ostensibly emanating from a “large, atomically smooth island” splits into a myriad of lines once the NSOM resolution is increased by bringing the tip close to the sample. NSOM images formed with a subset of these fine lines reveal a rich spectrum of interfacial roughness. Thus the structure of even the most perfect semiconductor interfaces is significantly more complex than suggested by the “atomically smooth island” interpretation of the PL data.

8.3.2.3 Interfaces Defined by Inhomogeneous Doping A “chemical interface” also results when a pn junction is formed by inhomogeneous doping of a semiconductor. Fundamentally, the silicon transistor, ubiquitous in ULSI *, is a highly inhomogeneous distribution of precisely placed dopants. The accurate fabrication of abrupt dopant “interfaces” on nanometer scales, and their control during subsequent processing steps represent key technological and scientific challenges. Until recently, however, no technique was available to reveal the distribution of dopants in materials in more than one dimension, Le., the depth below the surface. (Adequate 1D cal-

’ Ultra large scale integration.

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ibrations are possible by SIMS and spreading resistance profiling of test structures.) In 1998, Rau et al. were able to determine 2D maps of the electrostatic potential distribution, and hence the distribution of dopants in the bulk of materials with spatial resolution. Such resolution was urgently required by technological imperatives (see, e.g., Diebold et al., 1996). In order to appreciate the state of the art prior to this development, it is worth reviewing the procedures used in the past to determine microscopic dopant distributions. First, the solid state processes used to fabricate the device, such as implants, anneals, etches, etc., were simulated to “fabricate a virtual device”. Next, the electrical characteristics of the virtual device were simulated and compared with measurements from actual devices. Iterative corrections were made to the initial model until agreement was reached. The actual dopant distribution was then assumed to be that giving the best agreement between the simulated and measured electrical characteristics. Such laborious procedure are still prevalent today, because 2D dopant mapping in the bulk of devices by electron holography is a recent breakthrough. The importance of having a direct method capable of mapping 2D dopant distributions has long been recognized. Scanning capacitance microscopy (SCM) has produced compelling images, which unfortunately depend sensitively on the tip and the applied voltage. SCM images are thus difficult to translate into dopant maps without substantial modeling (Kleiman et al., 1997). Now, electron holography, an interferometric TEM technique, can be used to quantitatively map the 2D electrostatic field distribution across pn junctions in CMOS3 transistor structures. Due to the relatively Complementary metal oxide silicon transistor.

475

small concentrations of dopants involved (typically ~m-~~0.002-0.2%), the determination of 2D distributions with high spatial resolution is a major challenge. Fortunately, small chemical concentration changes result in strong variations of the local electrostatic potential at the pn junctions. The electrostatic potential is in fact the fundamental parameter needed for device modeling applications. In TEM, the local variations of the projected potential shift the phase of the high energy electrons used to illuminate the samples. In conventional TEM, this effect, and hence the influence of the electrostatic potential, are not visible. Electron holography, however, is able to measure the phase of the illuminating beam, and image the phase changes across the sample. Here, we provide a short introduction to the technique and describe how it can map the electrostatic potential across pn junctions inside CMOS transistor structures. The 2D dopant distribution is then determined by matching its corresponding potential distribution to the measurement. The basic principle of electron holography is shown schematically in Fig. 8-15. A thin, electron transparent sample is illuminated with a plane electron wave. A varying electrostatic potential distribution modulates the local phase of the electron wave. For a sample including apn junction viewed in cross-section under conditions that minimize dynamical diffraction effects (see, e.g., Spence, 1988), the phase shift is directly proportional to the electrostatic potential distribution AV,, across the junction (Frabboni et al., 1987; McCartney et al., 1994). Using an electron biprism, the modulated wave interferes coherently with a reference wave, which has passed through vacuum only. By Fourier analysis of the recorded electron hologram, the amplitude and phase of the modulated wave can be extracted (Lichte, 1997; Tonomura, 1987). The phase im-

476

8 Interfaces

Plane electron wave Reference wave

Sample,

cp = CEVo t

Modulated wave Objective lens Electron biprism fitted waves

Electron hologram

Figure 8-15. Principle of (off-axis) electron holography in TEM. A plane electron wave passing through the sample suffers a phase shift due to the electrostatic potential distribution in the sample. Using an electron biprism, the modulated wave is coherently interfered with a plane reference wave. Fourier analysis of the recorded electron hologram yields the amplitude and phase of the wave emerging from the sample.

Fourier analysis

age of a pn junction can be directly interpreted as a map of the projected electrostatic potential distribution. Figure 8-16 shows an electron hologram of a silicon test sample containing 20 nm wide boron marker layers at a depth of 100 nm and 300 nm from the silicon surface. The samples were grown by CVD with a boron doping level of 4 x 1019cm-3 on a phosphorus background

doping of 2x lo'* crnp3, as confirmed by SIMS measurements. This produces a potential change of l .05 V across the pn junctions. The change occurs over a depletion layer 27 nm wide. The electrostatic potential distribution across the pn junctions is captured as an additional bending of the interference fringes. Figure 8-17 a shows the phase image recon-

Figure 8-16. Hologram of a sample containing two boron marker layers. The positions of the layers are indicated. The hologram fringes act as a carrier frequency for the amplitude and phase of the electron image wave.


477

Figure 8-17. a) Phase and b) amplitude image reconstructed from the hologram in Fig. 8-16. The boron marker layers are clearly visible in the phase image. From the line scan across the phase image, a depletion width of about 25-27 nm across the pn junction is measured.

structed from the hologram. The potential drop across the marker layers is clearly revealed, and is in good agreement with the expected depletion layer width. The linescan indicates a spatial resolution of 5 nm. The amplitude image in Fig. 8-17b, corresponding to aconventional TEM image, cannot delineate the position of the marker layers. We now show that holography can be used to map the source and drain areas in submicrometer n- and p-channel transistor structures. Figure 8-18 shows amplitude and phase images of cross-sectional samples of 0.3 pm CMOS (complementary metal oxide semiconductor) transistors. The source and drain areas are clearly visible in the phase images and, as expected, show contrast reversal between NMOS (arsenic doped) and PMOS (boron-doped) devices. Most importantly, the phase images can be interpreted

directly, without simulation, and compared with amplitude images that correspond to conventional TEM imaging. The amplitude images show residual defects near the original wafer surface resulting from ion implantation. The location of such defects can therefore be precisely measured with respect to the extension of the pn junctions by combining amplitude and phase information. The phase images can be quantified using calibrated values for the electron-optical refractive index of silicon and by taking surface depletion effects on the top and bottom of the thin TEM samples into account '. Figures 8-19a and b are the amplitude and phase images of a 0.18 pm PMOS (buried channel) transistor. From the phase image, Details of the calibration procedure will be published elsewhere (Rau et al., 1999).

478

8 Interfaces

Fig. 8-18. Amplitude and phase images of 0.3 pm NMOS (left) and PMOS [right) transistors, viewed in cross section. The source and drain areas (marked as n+ or p') are clearly visible in the phase images with the appropriate contrast. Abrupt black-white contrast lines are due to phase changes larger then 2 x. The location of defects near the original wafer surface can be measured with respect to the extension of the source/drain areas.

Figure 8-19. Top: a) Amplitude image and b) phase image of a 0.18 pm buried channel PMOS transistor. Bottom: Extracted 2D potential distribution from the phase image in b). The sensitivity for measuring the electrostatic potential distribution across the pn junctions is 0.1 V.


we have extracted the depletion region potential distribution shown in Fig. 8-19c. Currently, we achieve a sensitivity of 0.1 V on measuring the electrostatic potential distribution across pn junctions (Rau et a]., 1998, 1999). The two-dimensional dopant profile is then determined by iteratively matching its associated potential distribution to the measurement. Such data can then be directly compared to the simulations of the fabrication process, thus eliminating the need to evaluate and calibrate process simulators by iterative matching of the electrical properties of macroscopic devices. In summary, electron holography can be used to image the electrostatic potential distribution across pn junctions in the bulk of materials and state-of-the-art devices. More generally, the approach opens the way for a 2D investigation of doping-related phenomena in semiconductor materials and devices with sub 10 nm resolution and 0.1 V sensitivity.

8.3.3 Relaxation of Chemical Interfaces Because semiconductor multilayers are becoming increasingly familiar, it is easy to overlook the fact that they are highly inhomogeneous systems far from equilibrium. On crossing a modern GaAdAlGaAs interface, the AI concentration changes by several orders of magnitude in a few lattice spacings. As pointed out by Cahn (1961), such systems relax by interdiffusion, sometimes in unusual ways. It is thus scientifically interesting and technologically important to investigate the stability of chemical interfaces against interdiffusion. In semiconductors, the modest diffusivities of point defects limit substantial relaxation at room temperature. However, an interface can relax during thermal annealing, in-diffusion of dopants, or ion-implantation. The extensive literature concerned with such phenomena

479

will not be summarized here, because excellent reviews already exist (Deppe, Holonyak, 1988). Rather, we describe the new understanding that emerges when the chemical relaxation of interfaces is studied at the atomic level. Using the quantitative chemical mapping techniques described above, it is straightforward to make sensitive measurements of interdiffusion at single interfaces. The composition profile across a given interface is measured in two pieces of the same sample, one of which has been annealed in bulk form (Fig. 8-20). Starting with the initial profile and using the diffusion coefficient D as free parameter, the diffusion equation is solved to fit the final (annealed) profile, thus deducing D as a function of temperature and interface depth (Kim et al. 1989, 1990a, b; Rouvibre et al., 1992). In the case of ion-implanted samples, the composition profiles are characterized by fitting an erfc-profile to the data, with the profile width L as the free parameter. Intermixing due to implantation is thus quantified in terms of changes in the interfacial width L (Bode et al., 1990).

-a s - g r o w n

r 0.4 c .c m 4L

W C

" c u 0

2 0 distance

-

Figure 8-20. Composition profiles of a C: GaAs/ AlGaAs interface at a depth of = 300 A,as-grown (solid line), and after 700°C for one hour anneal (dotted line). One standard deviation error bars are shown. Each measurement refers to a single atomic plane, and is obtained by averaging the composition over = 3 0 x 2 3 A' segments of an individual atomic plane.

480

8 Interfaces

8.3.3.1 Interdiffusion due to Thermal Annealing Fig. 8-21 is an Arrhenius plot of D vs. I/kT for C-doped GaAs/A10,,,Gao,63As interfaces at three different depths beneath the surface. Each measurement is made i n a cm3 in volume. Remarkably, region = the magnitude of the interdiffusion coefficient, and the activation energy for intermixing change strongly with depth (Rouvikre et al., 1992). Since this behavior is observed both in the GaAs/AlGaAs and the HgCdTel CdTe systems (Kim et al., 1989, 1990a, b; Guido et al., 1989), i t is likely that the depthdependence of the interdiffusion coefficient is a general effect. This is more clearly displayed in Fig. 8-22, where In D is plotted as a function of the interface depth. At the lower temperatures (700°C and particularly at 650°C), 1nD initially decreases linearly with increasing distance from the surface, but appears to drop exponentially beyond a certain critical depth.

T f°Cl 10-n

I

600

700

800 I

I

I

---_depth :

?

I

300

------- depth : 1500 A -depth : 2800 A

10-2'

I

I

1.0

11

1.2

1/ T(1P / K )

Figure 8-21. Arrhenius plot of the interdiffusion coefficient D at C: GaAslAlGaAs interfaces at three different depths.

c B

2 Y-

10-20

-

'c 'p L

0,

c

0

1000 1500 2000 2500 Interface Depth (A)

500

3 IO

Figure 8-22. Plot of the In D vs. interface depth z at three different temperatures, for C: GaAsIAlGaAs interfaces.

This has been shown to be related to the injection of point defects from the sample surface during the anneal. In particular, interdiffusion in these systems is assisted by the presence of native point defects (interstitials or vacancies), whose concentrations are often negligible in as-grown samples. For interdiffusion to occur, such native defects must be injected from the sample surface during the anneal. The interdiffusion coefficient is a sensitive function of the concentration of these defects at the particular interface studied, and thus can be used to investigate the microscopics of native point defect diffusion in multilayered systems. Indeed, it should be possible to measure the formation energy and migration energy of a given native defect (interstitial or vacancy) as a function of its charge state. Returning to interdiffusion, two important points emerge. First, the interdiffusion coefficient varies strongly with depth. Thus a measurement of this parameter is meaningful only if it refers to a single interface at a known depth. Second, it follows that the interface stability is also depth-dependent. Thus the layer depth must be regarded as an important design parameter in the fabrica-


tion of modem devices. This effect assumes additional importance when interdiffusion is also concentration dependent, leading to strong intermixing at very low temperatures (Kim et al., 1990a). These phenomena highlight the importance of a microscopic understanding of the relaxation of systems far from equilibrium.

8.3.3.2 Intermixing due to Ion-impIantation The passage of energetic particles through an inhomogeneous solid deposits sufficient energy in the solid to cause intermixing, even at very low temperatures. Using chemical mapping techniques, it is possible to detect the intermixing due to thepassage of a single energetic ion. Consider a GaAs/AlAs multilayer, held at 77 K and implanted with 320 keV Ga+ ions to a dose of 5 x lo'* cm-*, i.e. about one ion per 2000 A area of each interface. Fig. 8-23 shows a chemical lattice image of an unimplanted, 50A thick GaAs layer be-

481

tween its two adjacent AlAs layers, together with the composition profiles for each interface. The GaAs layer is situated 1400 p\ beneath the surface, and is thus close to the depth where the maximum damage during subsequent implantation is expected to occur. The growth direction is from bottom to top, the (later) implantation direction from top to bottom. Each point on the profiles of Fig. 8-23 represents the average composition of a 1 pm segment of a given atomic plane before implantation. Both top and bottom interfaces (A and B) display excellent lateral uniformity, and can be characterized by similar characteristic widthsL (LA= 2.4 k 0.1 A, LB=2.7kO.1 A). After implantation to a dose of 5 x IO'* cm-2, chemical analysis of individual interfaces located at depths between 1000 and 1700 A beneath the surface reveals significant intermixing across the top interfaces, although (on average) only one Ga+ ion has passed through each 2000A2 area of the interface. The intermixing is not uniform along the top interface, but shows large fluc-

Figure 8-23. Chemical lattice image of GaAs layer between two AlAs layers, with composition profiles across the interfaces. Growth direction is from bottom to top. One standard deviation error bars are shown.

482

8 Interfaces

Figure 8-24. Chemical lattice image of GaAs/AIAs sample implanted with 320 keV Ga’ ions to a dose of 5 x 10” cm-*. This dose corresponds to the implantation of =one ion per 45 A length of the interface. The composition profiles refer to adjacent 4 0 A segments of the top interface. Note the large local variations in intermixing on the 50 8, lateral scale. This correspond to the width of the damage track expected from the passage of a single ion.

tuations. In Fig. 8-24 three concentration profiles for adjacent 50 A segments of an interface are shown. Profiles a and c display similar degrees of intermixing, characterized by a width L of = 6.2 2 0 . 2 A,while profile b is characterized by L = 4 . 6 ? 0 . 2 A (Bode et al., 1990). Kinematic implantation simulations using the TRIM program (Biersack, 1987) show, that under the conditions used in these experiments, a single implanted Ga’ ion creates a damage track =50 A wide. This agrees closely with the width of the observed fluctuations in the degree of intermixing along the interface. After implantation at a higher dose 1 x l O I 3 cm-’, the intermixing along the interface is uniform. It is thus likely that the intermixing caused by the passage of single energetic ions is being directly imaged. In these experiments, a series of chemical interfaces is used as a stack of photographic emulsion layers, to record the passage of energetic ions, or native point defects. implanted at or injected from the surface. Thus chemical interfaces can be used

to reveal the microscopics of defect processes at the atomic level.

8.3.4 Summary At present, interfaces of the highest perfection, and thus widest application are those between lattice-matched, pseudomorphic, crystalline solids, differing only in composition. In Sec. 8.3, we attempted to outline the concepts needed to define such “chemical” interfaces. Two concepts emerge as fundamentally important. First, the definition of an interface is most conveniently affected in terms of its roughness spectrum, where the amplitudes of the interfacial undulations are specified as a function of their spatial frequency. Second, when one of two materials forming the interface is a random alloy, the interfacial configuration cannot be defined at the atomic level. Only when the length scale for the definition of the interface is so large that random alloy fluctuations are at a sufficiently low level, can an interface be adequately defined.

8.4 Interfaces Between Lattice-Mismatched, lsostructural Systems

Experimentally, it is essential to realize that any technique probes only a small part of the roughness spectrum. This “window” is delimited on the high frequency side by the spatial resolution of the technique, and on the low frequency side by the field of view. Moreover, a technique may possess an intrinsic length scale, the exciton diameter for luminescence, the Fermi wavelength of transport, which plays a crucial role in determining the wavelength of the interfacial roughness that can be most sensitivelyprobed. To gain a realistic impression of the interfacial configuration, information over a wide range of frequencies is needed. It is thus necessary to collate the data produced by a variety of techniques. The description of an interface in terms of an “island size” is an attempt to replace the real roughness spectrum essentially by a single frequency component. This is symptomatic of a simplistic interpretation of the data produced by any single experimental technique, and is too naive to be realistic. Chemical interfaces and multilayers are systems far from equilibrium, able to relax through interaction with point defects. This allows them to be modified by suitable processing for device applications. Equally importantly, chemical interfaces can be used to track the passage of point defects, providing a microscopic view of the processes that govern the elementary structural excitations of solids.

8.4 Interfaces Between LatticeMismatched, Isostructural Systems Although it is now possible to grow highly perfect semiconductor heterostructures where the constituent materials have identical, or very similar, lattice parameters, the ability to grow high quality heterostructures

483

with materials of significantly different lattice parameters is still evolving. Such lattice-mismatched heterostructures can exist either in a state where the lattice-mismatch strain is fully or partially relieved, or where no strain has been relieved. The primary structural issues are the presence of strainrelieving defects, known as misfit dislocations, and the planarity of the heteroepitaxial layers. In the following sections we will summarize the progress to date in understanding and controlling lattice-mismatched semiconductor heteroepitaxy.

8.4.1 Lattice Mismatch Strain and Relaxation Mechanisms 8.4.1.1 Origin and Magnitude of Lattice Mismatch Strain For simplicity, we consider heteroepitaxial systems comprised of constituent materials with cubic symmetry. An epitaxial layer can be grown upon a substrate with a different lattice parameter such that below a critical epilayer thickness, h,, the epitaxial material adopts the lattice parameter of the substrate parallel to the interfacial plane. This results in a biaxial interfacial strain of magnitude

(8-4) a,

where a, and a, are the relaxed (i.e., bulk) lattice parameters of the epilayer and substrate, respectively. By classical elasticity theory, this causes a tetragonal distortion of the unit cell, such that the lattice parameter normal to the interfacial plane aenis expanded (contracted) relative to the lattice parameter parallel to the interface plane, aepby:

sen= I + E,

U+V)

(8-5) (1 - v ) for epitaxial layers that have larger (smaller) lattice parameters than the substrate (vis the epilayer Poisson ratio). aeP

484

8 Interfaces

The lattice mismatch strain causes a biaxial stress (assuming an “infinite” or rigid substrate) within the epitaxial layer of magnitude 0,

=

2 E, G (1 + v) (1 - v )

Here G is the epilayer shear modulus. For lattice-mismatched systems of demonstrated or potential technological application, typical lattice mismatch strains are in the region of parts per thousand to parts per hundred. Thus for Ge,Si l-x/Si heterostructures, go= 0.041 x ; for In,Ga,-,As/ GaAs heterostructures, E, = 0.072 x ; and for In,yGa,,As/InP systems, E,= 0.072 (x-0.53). Growth of such alloy systems allows the strain to be tuned via the alloy composition, and thus to be kept low enough (typically 10.01) such that large densities of lattice mismatch dislocations are not generated. For larger mismatch systems, such as GaAs/Si (~,=0.041),InP/Si(~,=0.072), and AlN/Al,O, (~,=0.133 for the most common configuration of [OOOl] AlN parallel to [OOOl] A120, and [2110] AlN parallel to [l io01 A120,), it is not possible to prevent larger densities of misfit dislocations from forming. Practical application of such structures thus either requires control and elimination of mismatch defects (as discussed in Sec. 8.4.4), or for the defects to be benign (opto)electronically, as appears to be the case for GaN-based light emitting diodes (LEDs). Typical values of Poisson’s ratio in semiconductors are of the order of 1/3. Thus the ratio of interfacial to normal strains given by Eq. (8-5) is about 2. Typical values of shear moduli for most semiconductors of interest are in the range of lolo- 10’’ Pa. This produces lattice mismatch stresses of the order of 1 0 ” ~Pa, from inspection of Eq. (8-6). Enormous stresses result, e.g., of the order of 1 GPa

for a lattice parameter difference of around 1 %.

Finally, an additional potential source of strain between different epitaxial layers is the differential thermal expansion coefficients. These thermal mismatch strains are given by E,,=ATAK

(8-7)

Here A K is the difference in the thermal expansion coefficients for the two materials, and AT is the temperature excursion experienced (e.g., from growth or annealing temperature to room temperature, or vice versa). For semiconductors, values of K are typically of the order of 10“ K-I. Thus thermal mismatch strains are of the order of for typical temperature excursions of hundreds of degrees Celsius. For a given heteroepitaxial system, this typically corresponds to less than 10%of the lattice mismatch strain, and is thus relatively trivial. Thermal mismatch strains can become significant in the later stages of strain relaxed epitaxy (where misfit dislocations have relieved all or part of the lattice mismatch strain at the growth or annealing temperature), where thermal mismatch strains may generate additional dislocations during cooling.

8.4.1.2 Strain Accommodation and Relief Mechanisms The generic mechanisms for the accommodation and relief of lattice mismatch strain are shown in Fig. 8-25. In Fig. 8-25 a, the lattice mismatch strain is accommodated entirely by tetragonal straining of the epitaxial layer, as discussed in the previous section. In Fig. 8-25b-d, lattice mismatch strain is relieved by (b) roughening of the epitaxial layer, (c) interdiffusion, and (d) misfit dislocations. Each of the mechanisms (b) - (d) will now be summarized.


a.

b.

L.

d.

Figure 8-25. Schematic illustration of mechanisms for the accommodation of lattice mismatch strain: a) Elastic distortion of the epitaxial layer, b) roughening of the epitaxial layer, c) interdiffusion, and d) plastic relaxation via misfit dislocations.

8.4.1.3 Epitaxial Layer Roughening Both lattice mismatch strain and chemical dissimilarities may drive roughening or clustering of an epitaxial layer. Heteroepitaxial systems generally prefer to minimize their interfacial area, as may be visualized simply by analogy to the equilibrium model of the contact angle for a liquid drop on a planar surface ~ s =vX e s + ~ e COS v

8

(8-8)

Here xsv,xes, and xevare respectively the substrate surface, the epilayer-substrate interface, and epilayer surface energies, respectively, and 8 is the contact angle of the epitaxial island with the substrate. This equation will only be identically satisfied for planar growth by the homoepitaxial case of xev=xSvand xes= 0. In the general heteroepitaxial case, the epilayer will roughen or

485

cluster as an epilayer surface is created in preference to an epilayer- substrate interface. An intermediate growth mode also exists (Stranski and Von Krastanow, 1939), whereby layer-by-layer growth occurs up to a certain thickness, and is then followed by island formation. This intermediate growth mode is widely observed in the Ge (Si)/Si system, for example. Figure 8-25 b illustrates how simple consideration of elastic constraints demonstrate that the formation of islands or undulations in an epitaxial layer can relax strain. This is because the peaks of the islands or undulations can relax towards the equilibrium lattice constant of the epilayer film. However, there is also an increase in constraint at the island edge or at the trough of the undulations, resulting in an increase in stress at these locations. For systems such as GexSi,,/Si and In,Ga,,As/GaAs, where the strain in the film is compressive, island formation allows for a tensile relaxation at the island peaks and an increase in compressive strain and stress at the island edge. This change in the distribution of strain results in a lowering of the total elastic strain energy of the system. This is because the volume of material underneath the relaxed peaks (which experiences stress relief) is greater than the volume of material in regions at island edges, which experiences increased compressive stress. Note that controlled understanding and engineering of strain in clustered epitaxy has also opened new avenues for the fabrication and “self-assembly” of quantum dot arrays, where each semiconductor island may be considered as an individual quantum dot, Le., it exhibits three-dimensional electron confinement. The primary goals in creating such structures are that the islands have a narrow size distribution and a high spatial density. The majority of the experimental work on self-assembled quantum

486

8 Interfaces

dots has been on the InAs/GaAs (e.g., Petroff and Medeiros-Ribera, 1996) and Ge/Si (e.g., Krishnamurthy et al., 1991; Teichert et al., 1996: Kamins et al., 1997) systems. Due to the intense interest in fabricating quantum dot arrays, the number of publications in this field is growing rapidly. It is found that improvements in island size uniformity and increases in spatial density may be obtained by growing multilayer structures. Transmission electron microscopy of Ge,Si,JSi (e.g., Kuan and Iyer, 1991) and InAs/GaAs (e.g., Xie et al., 1995a) multilayer structures has shown that islands in successive layers can exhibit a high degree of spatial correlation, with islands aligning themselves above previously formed islands i n the lower layers. This behavior is due primarily to the presence of localized areas of reduced misfit strain above previously existing islands, causing preferential nucleation of new islands (Teichert et al., 1996; Tersoff et al., 1996). Theoretical descriptions of strain relief by island formation and surface roughening have generally used either continuum (e.g., Mullins, 1957; Srolovitz, 1989; Grinfeld, 1993; Spencer et al., 1991) or atomistic approaches. The continuum models consider the interplay between the relief of elastic strain energy and the increase in surface energy caused by diffusion-driven surface roughening. The atomistic approaches focus upon the roles of surface energies of island facets, and the interaction of steps during island formation (e.g., Tersoff and LeGoues, 1994; Tersoff and Tromp, 1994; Xie et al., 1995b; Chen et al., 1997; Jesson, 1998). Both these viewpoints yield valuable insight into the fundamental mechanisms involved in epitaxial roughening during both growth and annealing of strained heterostructures. However, there is as yet no single theoretical treatment capable of describing the entire range of observed island-

forming and roughening behavior as concisely as the Matthews-Blakeslee (19741976) and Dodson-Tsao (1987) models do for strain relief by misfit dislocations (see Sec. 8.4.2). In practical applications, the tendency to clustering should be eliminated or minimized for semiconductor device applications. This may be achieved by: (i) Reducing chemical dissimilarity across the interface, e.g., for an alloy A,B,, grown on B, reducing x . This reduces the interfacial energy xes. (ii) Reducing the growth temperature, as this has the effect of reducing the surface mobility of deposited atoms, thereby preventing them from achieving their equilibrium state. Clustered growth is therefore effectively “frozen out”. (iii) Reducing the lattice mismatch, as clustering is also driven by strain relief.

The above trends are illustrated experimentally in Fig. 8-26, which shows measured regimes of clustered vs. layerby-layer growth in the Ge,Si,,/Si (100) system (Bean et al., 1984).

Ge,Sil-,

800

Morphology cl

0

I

Three Dimensional Growth (Is1anding)

T w o Dimensional Growth (Planar)

LOO

0

0

I

I

0 l

I

1

0 1

1

1

1

0.1 0.2 0 . 3 0 4 0.5 0.6 0.7 0.8 0.9 1.0 Germanium Fraction (X)

Figure 8-26. Regimes of layer-by-layer vs. islanded growth in Ge,Si,-,lSe(lOO) (Bean et al., 1984).


Lattice mismatch relief mechanisms are significantly different in clustered growth as opposed to planar growth. Only a brief description of these differences will be given here, as the goal in practical heteroepitaxy is to suppress clustering. However, some important heterointerfaces do involve clustered growth at early stages of the growth process, e.g., in the GaAs/Si system. In homoepitaxy, GaAs is generally grown at 600 “C. However, growth of GaAs on a silicon substrate at this temperature leads to clustered growth, which persists up to relatively thick (hundreds of nm) layers. Clustering is thus generally suppressed in the initial stages of heteroepitaxy on the silicon substrate by using an unusually low GaAs growth temperature 300-400°C. This reduces surface diffusion, producing a higher density of smaller nuclei than would be expected at higher temperatures (Biegelsen et al., 1987). By the time a layer thickness of the order of 100 nm is reached, the GaAs layer, although not absolutely planar, is at least continuous with no bare silicon substrate remaining (Harris et al., 1987). Subsequent higher temperature (Le., in the normal homoepitaxial deposition regime) GaAs growth causes the layer to planarize, as homoepitaxial growth conditions have now effectively been established. The GaAs/Si cluster interfaces are initially coherently strained (Hull and FischerColbrie, 1987j.As the island dimensions get larger, the interfaces dislocate. This occurs at a critical transition which depends upon all the island dimensions. In practicular, for island widths not much greater than the equilibrium interdislocation spacing required to fully relax strain in an equivalent planar structure, the interface remains coherent for island heights substantially greater than the equivalent planar critical thickness (Luryi and Suhir, 1986; Hull and Fischer-Colbrie, 1987). Another phenome-

-

-

487

non driving increased critical thickness in clustered structures is elastic distortion of the underlying substrate, which effectively equipartitions the strain between the substrate and the strained cluster (Eaglesham and Cerullo, 1990). In summary of this section, due to lattice strain and interfacial energies, clustered growth may be regarded as the general heteroepitaxial growth mode, although techniques for encouraging layer-by-layer growth exist. The presence of the cluster surfaces modifies the energetic analyses for misfit dislocation introduction, in general allowing dislocation free islands to be grown to greater “critical thicknesses” than equivalent planar structures. 8.4.1.4 Interdiffusion

The elastic strain energy stored in an epitaxial layer is proportional to the square of the elastic strain. Thus for an epitaxial layer that is miscible with the substrate (e.g., Ge,Si,,/Si, In,Ga,,As/GaAs), the strain energy may be relaxed by interdiffusion. To illustrate this, consider an alloy layer AXfB of thickness ho grown upon a substrate of material B, where the lattice-mismatch strain For an initially abrupt interface, the is Cox’. elastic strain energy Eo= k’ ho( Cox’)2,where k’= 2 G ( 1 + Y)/(1- vj. Following a thermal cycle, the element B diffuses into the substrate, reaching a distance hf from the surface. k’ CiJx2( h ) The new strain energy is dh, integrated between h=O and h = h f . For h f > h o ,hOxt2>.fx2(hjdh for all physically plausible forms of x (h).For example, in the (physically unlikely) event that the germanium redistributes itself uniformly within the new layer depth of hf, Jx2(h )dh =xt2hi/hf. Significant interdiffusion is unlikely, however, at typical growth, annealing, or processing temperatures in the most commonly studied systems. For example in the

,-,‘

488

8 Interfaces

Ge,Si,-JSi system, diffusion lengths of one monolayer would take the order of 1 h at 800°C (e.g.,Fiory et al., 1985; Van de Walle et al. 1990). One configuration where interdiffusion can be significant, however, is in ultra-short (or the order a few monolayers) period superlattices with high strain (e.g., pure germaniudpure silicon), where intermixing at the interfaces can cause significant interdiffusion for timehemperature cycles as low as tens of seconds at 700°C (e.g., Lockwood et al., 1992; Baribeau, 1993).

8.4.1.5 Misfit Dislocations As shown in Fig. 8-25d, an interfacial dislocation array can allow the epitaxial layer to relax towards its bulk lattice parameter. The driving force for the introduction of this dislocation array is relaxation of the epilayer elastic strain energy. Balancing this energy gain, however, is an increase in energy due to the dislocation self-energy, arising from electronic and nonelastic distortions at the core, and the elastic strain

field around the dislocation core. The net energy change in the system from introduction of the dislocation array is thus negative only for epitaxial layer thicknesses greater than a minimum value, known as the critical thickness h,. Quantitative expressions for this parameter will be derived in Sec. 8.4.2. It is a geometrical property of a dislocation that it cannot terminate in the bulk of a crystal, but only at a free surface, at an interface with noncrystalline material, at a node with another dislocation, or by forming a complete loop within itself. Thus the strainrelieving interfacial segment, known as the misfit dislocation, must necessarily be terminated at its ends by two threading dislocations propagating to the free surface. This is illustrated in Fig. 8-27. The exceptions to this rule will occur if the interfacial segments extend to be long enough such that they extend to the edge of the substrate wafer, or if the dislocation density is high enough such that dislocation interactions and reactions occur in the interfacial plane.

Figure 8-27. Schematic illustration of the geometry of a misfit dislocation (BC), and accompanying threading dislocations (AB and CD). Also shown are the four possible Burgers vector orientations for a given misfit dislocation and glide plane.


Most widely studied semiconductor heterostructures (e.g., germanium, silicon, most III-V compounds, and alloys thereof) are comprised of constituents with lattice structures that are diamond cubic (dc) or zincblende (zb). In these systems, as in the parent face centered cubic (fcc) structure, the preferred glide plane for dislocations is the { 11 1 } plane. Thus the orientation of interfacial dislocation arrays is determined by the intersection of inclined { 11 1 } glide planes with the interfacial plane. As illustrated by Fig. 8-28, this yields orthogonal, hexagonal, and linear arrays for (100) and (1 1 1) and (1 10) interfaces, respectively, with dislocations lying along in-plane (01 1) directions. The preferred Burgers vector of perfect dislocations in the fcc, zb, and dc systems is b=a/2 (llO), as this is the minimum lattice translation vector in the system. As shown in Fig. 8-27, a given interfacial dislocation may have four different Burgers vectors satisfying this relation. Of these, one ( b , in Fig. 8-27) is in the screw geometry and experiences no lattice-mismatch force [as may be seen from inspection of Eq. (8-9)]. Two (b2and b, in Fig. 8-27) lie within the inclined glide plane, and dislocations with these Burgers vectors can grow rapidly by glide. The angle, 8, between the interfacial dislocation line direction and the Burgers vectors is 60". The final orientation (b4 in Fig. 8-27) corresponds to a dislocation with

100

110

..'

1

111

.

Figure 8-28. Schematic illustrations of the symmetries of interfacial misfit dislocations at (loo), ( I lo), and (1 11) interfaces in f.c.c., d.c., or z.b. systems with intersecting ( 11 1 } glide planes.

489

a Burgers vector lying out of the glide plane; such dislocation are pure edge in character ( 8 = 90") and can only move by far slower climb processes. It is also the case that on the nanometer scale the total b =a/2 (1 IO) dislocations dissociate into b = a / 6 (211) Shockley partial dislocations, separated by a ribbon of stacking fault. The equilibrium zero-stress dissociation width in most bulk semiconductors (e.g., silicon, germanium, GaAs) is of the order of a few nanometers, as the stacking fault energy is relatively high [typically in the range 30-70 mJ m-* (George and Rabier, 1987; Hull et al., 1993)l. The lattice mismatch stress can act so as to compress or separate these partial dislocations from their equilibrium dissociation width. For the most common configuration of a compressively strained epilayer (e.g., Ge,Si l-x/Si, In,Ga,,/As) with a (100) interface, the lattice mismatch stress compresses the partials together and the total defect can be approximatedas an undissociatedb=a/2 (1Ol)dislocation. For other configurations [such as tensile strained epilayers on (100) interfaces, or compressively strained epilayers on (1 10) or (1 11) interfaces], the lattice mismatch stress acts so as to separate the partials. For some combinations of epilayer thickness and strain, this separation may be infinite, and then the operative misfit dislocation becomes of the b = a/6 (21 1) type. For detailed analyses of this phenomenon, see Hull et al. ( 1 992, 1993).

8.4.1.6 Competition Between Different Relaxation Modes Of the strain relief mechanisms discussed in the preceding four sections, interdiffusion is significant only at growth or annealing temperaturehime cycles of the order of 8OO0C/1 h or greater (except for the relatively specialized configuration of mono-

490

8 Interfaces

layer scale superlattice structures). Thus, interdiffusion is not a significant mechanism at the growth temperatures typically used during MBE or UHV-CVD growth, although it could conceivably be significant during high temperature post-growth processing (e.g., implant activation or oxidation processes). Relaxation via surface roughening can occur for any epitaxial layer thickness, providing the chemical dissimilarity, strain, and temperature are sufficiently high. Strain relaxation by misfit dislocations occurs only for layer thicknesses greater than the critical thickness. Both processes are kinetically limited, either by surface diffusion lengths in the case of surface roughening, or by dislocation nucleation/propagation barriers in the case of misfit dislocations. These processes will be competitive if the strain relaxed by surface roughening reduces or eliminates the driving force for dislocation introduction. They will cooperative if the morphology generated by roughening reduces energetic barriers for dislocation nucleation (e.g., Cullis et al., 1994; Jesson et al., 1993, 1995; Tersoff and LeGoues, 1994), or the strain fields associated with misfit dislocations induce surface morphology (e.g., Hsu et al., 1994; Fitzgerald and Samavedam, 1997). The detailed balance between roughening and dislocation generation is still a topic of active experimental research and simulation. Regimes of temperature, epilayer thickness, and strain where surface roughening or misfit dislocations dominate have been mapped for the Ge,Si,,/Si system (Bean et al., 1984; Lafontaine et al., 1996: Perovic, 1997).

8.4.2 The Critical Thickness for Misfit Dislocation Introduction: Excess Stress 8.4.2.1 Basic Concepts: Single Interface Systems The critical thickness for misfit dislocation introduction may most conveniently be formulated by considering the stresses acting on a misfit dislocation. This approach is directly analogous to the original MatthewsBlakeslee (1974- 1976) “force balance” approach. The resolved lattice mismatch stress acting on a misfit dislocation is

o,= a, s=cr, C O S L cos$

(8-9)

Here a, is given by Eq. (8-6), A is the angle between b and that direction in the epilayerhubstrate interface that is perpendicular to the misfit dislocation line direction, and @ is the angle between the glide plane and the interface normal. The quantity S is called the Schmid factor, and resolves the lattice mismatch force onto the dislocation. The lattice mismatch stress drives growth of the misfit dislocation, thereby relaxing elastic strain energy in the epitaxial layer. Balancing that is a “line tension” stress resulting from the self energy of the dislocation. Standard dislocation theory (e.g., Hirth and Lothe, 1982) gives for this stress Gbcos$(1-vcos26)) ]lnTa h (8-10) 4 Jt h (1 - v ) Here CY is a factor accounting for the dislocation core energy. The net or “excess” stress (Dodson and Tsao, 1987) acting on a perfect dislocation is then

a,, = a, - (31.

(8-11)

If this excess is greater than zero, then growth of misfit dislocations is energetically favored; if it is negative, then misfit dislocations are not energetically favored. The


equilibrium state of the system is that the excess stress is zero for any intermediate strain state during relaxation of the system. The critical thickness, h,, is found by solving for h at o,, = 0 and E = E,, yielding

b (l-v cos2 6 ) In (ahc l b )

(8-12) [8 (1 +v) E cosai Note that comparable expressions have been derived by energy minimization approaches (e.g., Van der Merwe and Ball, 1975; Willis et ai., 1990). In addition, several refinements of the force-balance model of Matthews-Blakeslee have been published, including incorporation of surface stresses (Cammarata and Sieradzki, 1989), consideration of the orientation of the threading arm within the glide plane (Chidambarrao et al., 1990), and calculations based upon anisotropic elasticity (Shintani and Fujita, 1994). Experimental measurements of the critical thickness in III-V compound semiconductor systems have generally exhibited relatively good agreement with the predictions of Eq. (8-12) (e.g., Fritz et al., 1985; Gourley et al., 1988; Temkin et al., 1989). This is because the Peierl's barrier for dislocation motion is relatively low in these materials, of the order of 1 eV (George and Rahc =

'0"atO

'

0:l 0.2 0:3 0.4 015 016 017 O:S?.9 '

'

'

'

'

X

'

'

491

bier, 1987), and thus the dislocation propagation kinetics are relatively rapid, allowing the system to relax rapidly at the growth temperature when the critical thickness is exceeded. In silicon-based systems, however, where the Peierl's barrier is higher (2.2 eV in silicon), relaxation kinetics are much more sluggish, and the critical thickness for dislocation motion may be significantly exceeded before detectable misfit dislocation densities are formed. This is illustrated by Fig. 8-29, where the predictions of Eq. (812) are compared with experimental measurements (Bean et al., 1984; Kasper et al., 1975; Green et al., 1991; Houghton et al., 1990) of critical thickness for different growth or annealing temperatures in the Ge,Si,,/Si (100) system. It is observed that at lower growth temperatures, the equilibrium critical thickness is significantly exceeded, whereas at higher temperatures Eq. (8-12) predicts the data relatively accurately. This is due to thermally activated kinetics for dislocation nucleation and propagation, as will be discussed in Sec. 8.4.3.

8.4.2.2 Extension to Multilayer Systems Understanding strain relaxation mechanisms at a single interface is of great impor-

110

Figure 8-29. Predictions of Eq. (8-12) ("equilibrium") for the critical thickness, h,, in Ge,Si,,/Si(100) structures, with the dislocation core energy parameter a= 2. Also shown are experimental measurements of h, for different growth/annealing temperatures from the work of a) Bean et al. (1984), b) Kasper et al. (1975), c) Green et al. (1991), and d) Houghton et al. (1 990).

492

8 Interfaces

tance, because these represent the simplest model system in which to study the fundamental misfit dislocation processes. However, strained layer geometries of practical importance to electronic devices generally involve more than one interface. For example, high speed heterojunction transistors such as modulation doped field effect transistors (MODFETs) or heterojunction bipolar transistors (HBTs) in their simplest form utilize the interface sequence A/B/A (with appropriate doping transitions), with B the heteroepitaxial strained layer. Thus misfit dislocation mechanisms in double and multiple interface systems are of great interest. In the simplest case of an A/B/A structure, it might be expected that growing dislocation loops should simultaneously relax the top B/A and bottom A/B interfaces. If only the bottom interface were relaxed, as in the single interface A/B case, then the B layer relaxes towards its own natural lattice parameter, but the top A layer would be forced to adopt this same value and would become strained. Thus, as illustrated in Fig. 8-30, both interfaces, generally relax simultaneously. This situation may be approximated by a simple modification to the Matthews-Blakeslee model

o,, - 0, - 2 (TT

(8- 13)

Here the factor of two before the C J ~term arises from the need to generate two misfit dislocation line lengths at the top and bottom interfaces. Only in the limiting case where the capping layer becomes very thin (say substantially thinner than the buried B layer) is the situation reached where relaxation of only the bottom interface becomes favorable (Tsao and Dodson, 1988; Twigg, 1990). This will occur if the strain energy involved in distorting the A capping layer is less than the self energy of the top dislocation line.

Figure 8-30. a) Schematic illustration and b) experimental verification by plan-view TEM of the prevalent misfit dislocation geometry in buried strained epitaxial layers. [Here illustrated for the Si/Ge,Si,_,/ Si(100) system.]

An equivalent strained B layer grown in the buried (A/B/A) configuration as opposed to the free surface (B/A) configuration will generally be more resistant to strain relaxation because:


(i) The Matthews-Blakeslee excess stress is smaller (ii) Dislocation nucleation is inhibited, as any generation within the strained layer requires full loop nucleation, as opposed to the half loop nucleation possible at a free surface. This greatly increases the activation barrier associated with loop nucleation (see Sec. 8.4.3.2). (iii) It has been observed that dislocation propagation velocities are lower in capped epilayers than in uncapped epilayers with equivalent excess stress (Hull et al., 1991a; Hull and Bean, 1993). This is attributed to different kink nucleation mechanisms in the two systems. The extra stability of buried layer structures is a great advantage in device processing, where the structure may have to be exposed to processing temperatures substantially greater than the original growth temperature. Significant increases in interfacial stability have been reported (e.g., Noble et al., 1989). With regards to extension to superlattice systems, it can be shown (Hull et al., 1986) that the equilibrium limit of the strained layer growth before misfit dislocation introduction in a superlattice A/B/A/B/A ... grown on a substrate A is equivalent to that of a single layer of the average superlattice strain, weighted over the thicknesses and elastic constants of the individual layers. The overall “superlattice critical thickness”, H, , may therefore be substantially greater than the single layer critical thickness h, particularly if the thickness of the A layers is substantially greater than the thickness of the B layers, and provided that each B layer is thinner than the appropriate critical thickness for B/A growth. If the superlattice thickness exceeds H, under these conditions, the relaxation of the superlattice oc-

493

curs primarily at the substrate/superlattice interface (Hull et al., 1986). If individual B layers exceed the critical thickness for B/A growth, then relaxation also occurs at intermediate interfaces within the superlattice. Next, consider a substrate of a different material C, with a lattice parameter corresponding to the average (weighted over layer thicknesses and elastic constants) of the superlattice layers A and B. Providing the individual layers of A and B do not exceed the A/C or B/C single layer critical thicknesses, arbitrarily thick superlattices may then be grown without the introduction of misfit dislocations. This has been demonstrated for ultra-thin (- 1 nm) GaAs-InAs superlattices on InP substrates (Tamargo et al., 1985). Although the lattice mismatches of GaAs and InAs to InP are -3.6% and + 3.6%, respectively, the average lattice parameters of GaAs and InAs are very close to that of InP, and many thin bi-layers (with equal GaAs and InAs layer thicknesses) may be grown without interfacial dislocations appearing. It should also be stressed that relaxation in strained layer superlattices may be very sluggish. As pointed out by several authors (e.g., Miles et al., 1988), misfit dislocations have to traverse many interfaces in these structures and their net velocities may thus be relatively low.

8.4.3 Misfit Dislocation Kinetics 8.4.3.1 Kinetic Relaxation Models Figure 8-29 highlights the importance of misfit dislocation kinetics in lattice mismatch strain relief, particularly in system such as Ge,Si,,/Si where the activation barriers for dislocation nucleation and motion are relatively high. Finite relaxation kinetics inhibit the growth of the dislocation array, and prevent the system from main= 0. taining equilibrium, defined as uex

494

8 Interfaces

A complete kinetic treatment of strain relaxation by misfit dislocations requires consideration of the processes of dislocation nucleation, dislocation propagation, and dislocation interactions. Each of these processes will be reviewed i n subsequent sections of this chapter. The first comprehensive kinetic description of relaxation by misfit dislocations in strained layer heterostructures was for the Ge,Si,,/Si system by Dodson and Tsao (1987). They combined the concepts of: (i) Excess stress, as defined by Eq. (8-11). (ii) Existing measurements of thermal activation parameters for dislocation propagation in bulk silicon [e.g., Alexander (1986), George and Rabier ( 1987), Imai and Sumino (1983)], and (iii) Dislocation nucleation from combination of multiplication and a pre-existing source density. They termed the resulting model a “theory of plastic flow”. Using this model, they were able to simulate a wide range of existing experimental measurements of strain relaxation by heterostrucmisfit dislocations in tures. Subsequently, Hull et al. (1989a)modeled the strain relaxation process in Ge,Si,-, based structures by direct measurement of misfit dislocation nucleation, propagation, and interaction rates using in situ TEM observations. These measurements were incorporated into the following description of the strain relaxation process [ L( t )b cosh] A&( t )=

--( -“ c y )

j~ ( tN) ( t ) dt

(8-14)

Here A&( t ) , L ( t ) , N ( t ) , and v ( t ) are the magnitude of plastic strain relaxation, total interfacial misfit dislocation length per unit area of interface, the number of growing dislocations, and the average dislocation velocity, respectively, at time t. The integral is

evaluated over all the time for which the excess stress is positive. Appropriate descriptions of N ( t ) and II ( t ) were determined experimentally, and misfit dislocation interactions were incorporated by reducingN ( t )by the number of dislocations pinned according to the discussion in Sec. 8.4.3.4. Subsequent models have developed Eq. (8-14). Houghton (1991) applied a version of Eq. (8-14) to the initial stages of relaxation, where dislocation interactions are relatively unimportant. The resulting expression developed by Houghton from direct measurements of dislocation nucleation and propagation rates (using combined etching and optical microscopy techniques) was 4.5(eV)

-de(r’

dt

- 1.9x104t N,(o,,)~.~e-kr

(8-15)

Here No is the initial source density of misfit dislocations at time t = O . This expression implies an enormous sensitivity of relaxation rate to both temperature and excess stress. Note also that Eq. (8-15), as reproduced here from Houghton (1991), effectively has a factor G-4.5incorporated into the prefactor to make it dimensionally correct. Gosling et al. (1994) subsequently developed the concepts of Eq. (8-14) using more complete descriptions of misfit dislocation interactions, and a fittable form for dislocation nucleation. They were able to successfully reproduce the experimental data presented in Hull et al. (1989a).

8.4.3.2 Nucleation of Misfit Dislocations A central and still somewhat controversial question in strained layer epitaxy is the nature of the source for the very high defect densities in relaxed films. Unless a sufficient density of nucleation sources exists, the interface will never be able to relax to its equilibrium state, no matter how long


each individual defect grows. Systems do exist in which relaxation is nucleation limited in this fashion. A simple calculation shows that to relieve a lattice mismatch of 1% across a 10 cm wafer requires lo6 m of dislocation line length. This requires a substrate defect density of lo5 cm-*, even if each dislocation grows sufficiently long to form a chord across the entire wafer. Matthews and Blakeslee (1974- 1976) originally assumed that these defects originated from exiting dislocations in the substrate wafer. Contemporary silicon and GaAs substrates have dislocation densities of 10’-lo2 and lo2-lo4 cm-2, respectively. Additional sources of misfit dislocations are thus clearly required. Three such general classes of misfit dislocation sources may operate in strained layer epitaxy: (i) multiplication mechanisms, (ii) homogeneous sources arising from the inherent strain in the epitaxial layer(s), and (iii) heterogeneous sources arising from growth or substrate nonuniformities. Multiplication mechanisms are particularly attractive as they significantly reduce the required density of dislocation sources and are generally invoked in plastic deformation in bulk semiconductors (Alexander, 1986; George and Rabier, 1987). The original example of such a multiplication mechanism in strained layer relaxation was proposed for the Ge/GaAs system by Hagen and Strunk (1978). In this model, intersections of orthogonal dislocations with equal Burgers vectors glide from the interface to the surface under the influence of image forces. Then, on intersecting the free surface they form new dislocation segments. This process can act repeatedly to produce bunches of parallel dislocations with equal Burgers vectors. This mechanism has since also been invoked in the InGaAs/ GaAs [e.g., Chang et al. (1 988)] and GeSi/Si [e.g., Rajan and Denhoff (1987)l systems.

-

-

-

-

495

Several additional multiplication mechanisms have been reported in the Ge,Si,,/Si system. Tuppen et al. (1989,1990) used Nomarski microscopy of defect-selective etched Ge,Si,,/Si( 100) structures to observe dislocation multiplication associated with dislocation intersections. Two distinct Frank-Read type and cross-slip mechanisms for the multiplication process were proposed based upon dislocation intersections acting as pinning points for dislocation segments. These pinned segments then subsequently grow by bowing between the pinning points and configuring into re-entrant and re-generative geometries similar to the classic Frank-Read source (Frank and Read, 1950). Capano (1992) described several multiplication configurations that did not require dislocation intersections, generally involving dislocation cross-slip following pinning of segments of the dislocation by inherent defects in the dislocation or host crystal. A Frank-Read type source in Ge,Si,,/Si epitaxy has also been described by LeGoues et al. (1991), who invoked dislocation interactions at the interface to provide the required pinning points. Most of these proposed multiplication mechanisms require an initial source density to provide the dislocations required to fuel the multiplication events arising from intersections. In addition, most configurations require a minimum epilayer thickness to accommodate the intermediate “bowing” configurations during the regenerative mechanism. The experiments of Tuppen et al. (1989, 1990) and Capano (1992), for example, established minimum epilayer thicknesses of about 0.7 pm for multiplication in the Geo.13Sio,87/Si( 100) system. By homogeneous nucleation, we mean sources not associated with any specific site or fault in the lattice, but rather with the inherent strain field in the epitaxial layer. If such sites exist, they would be present in

496

8 Interfaces

very high densities, providing a very high prefactor in an Arrhenius equation. This prefactor may be estimated by (8-16) Here vo is an attempt frequency for critical dislocation nucleus formation, which might be approximated by the Debye frequency (- 10l3 s-’ in silicon), N , is the number of atoms per unit area in the epitaxial layer (- l O I 5 cm-’), and Ncritis the number of atoms in the critical nucleus projected onto the epilayer surface (- lo3 for a strain of 0.01, as discussed in the analysis below). This produces a prefactor of the order of cm-’ s-l. The process will be thermally activated according to the standard Arrhenius relation (8-17) Thus to obtain a measurable nucleation rate (say > 1 cm-’ s-’) will require an activation energy of the order of 5 eV or less at a temperature of 600°C. The energetics of nucleation of complete dislocation loops within a layer, or half loops at the free surface, has been discussed by a number of authors [e.g., Eaglesham et al. ( 1 989), Fitzgerald et al. (1989), Hull and Bean ( 1989a), Kamat and Hirth ( 1990), Matthews et al. (1976), Perovic and Houghton (1992)l. The latter case of surface nucleation is generally expected to dominate due to the lower line length (and hence self energy) of a half versus a full loop. The total system energy is calculated by balancing the dislocation self energy with strain energy relaxed and the energy of surface steps created or destroyed as a function of the loop radius R €total= €loop-

Estrain

= A R lnR-BR’+CR

* Estep

(8-18 a) (8-18b)

Here, A, B and C are compound elastic and geometrical constants. The dislocation self-energy, Eloop,varies as R In R and dominates at low R, while the strain energy relaxed, Estrain, varies as R’ and thus dominates at high R. As illustrated in Fig. 8-3 1, the passes through a total system energy, Etotal, maximum, 6E, at a critical loop radius R, and then monotonically decreases. The quantity 6 E thus represents an activation barrier to loop nucleation, and depending upon the elastic constants of the particular system under consideration, is typically very high (tens to hundreds of electron Volts) for strains 0.7 eV on ptype silicon) measured from uniform Nisi, layers is also very different from the value of 0.6-0.7 eV usually observed for all phases of polycrystalline nickel silicides on silicon. As shown in Fig. 8-40, the facet bar density at an NiSi,/Si (1 00) interface can be controlled by processing. The presence of a few facet bars at NiSi,/Si (100) interfaces, which are otherwise flat, has little effect on the n-type SBH, but has a strong influence on the measured SBH of p-type silicon. The I - V deduced p-type SBH decreases rapidly as the density of facet bars increases, while a slower but noticeable decrease of the C- V SBH is concurrently observed. As a result, the C-V measured SBH for any specific diode significantly exceeds that deduced from I - V. Mixed-morphology p-type diodes are “leaky”, having poor ideality factors ( n 2 1.08 for N , > 10l6 ~ m - and ~ ) displaying reverse currents that do not saturate. There is also a clear dependence of the electron transport on the substrate doping level. All of these observations are consistent with the behavior of an inhomogeneous Schottky barrier, as explained in detail in an analytic theory on SB inhomogeneity (Tung, 1991; Sullivan et al., 1991). If it is assumed that the faceted regions have a local SBH of 0.47 eV, characteristic of the sevenfold type A ( 1 1 1 ) interface, while the planar background has a SBH of 0.72 eV, characteristic of the Nisi,( 100) interface, then the experimental results can even be explained semiquantitatively. The epitaxial system of NiSi,/Si( 100) presents a very rare opportunity i n the study of the SB formation mechanism, because it is the only demonstration of SBH inhomogeneity, on a length scale

8.5 Interfaces Between Crystalline Systems Differing in Composition and Structure

smaller than the depletion width, which is artificially fabricated and studied. Results from this interface provided strong support from the transport theory of inhomogeneous SBs (Tung, 1991). The dependence of the SBH on interface orientation, with three different SBHs for three differently structured interfaces between the same two materials, has not been observed for nonepitaxial M-S systems and appears to be in disagreement with many existing SBH theories. Theoretical calculations have shed some light on the origin of the observed difference in A and B type SBHs. Very large supercell sizes were used and were found to be necessary to observe the difference in interface electronic properties due to the subtle difference in the interface atomic structures (Hamann, 1988b; Das et al., 1989; Fujitani and Asano, 1989). The experimentally observed difference in A and B type SBH is qualitatively reproduced in these first-principle calculations. Quantitative agreement also seems reasonable with the most sophisticated calculations. At this stage, it seems appropriate to attribute the mechanism for SBH formation, at least for these near perfect M-S interfaces, to intrinsic properties associated with the particular interfacial atomic structure. The SBH at the type B CoSi,/Si( 111) interface is also expected to depend on the interface atomic structure (Rees and Matthai, 1988). Films grown by room temperature deposition of 1-5 nm thick cobalt and annealing to > 600 "C usually show an SBH in the range 0.65-0.70 eV on n-type silicon (Rosencher et al., 1985). These interfaces have the eightfold structure. However, recent experiments on the growth of CoSi, layers at lower temperatures and with lower dislocation densities have produced interfaces that show considerable variation in SBH (Sullivan et al., 1993). A low SBH, < 0.3 eV, for some type B CoSi, layers on n-

-

523

type Si (1 11) has been observed, which has tentatively been attributed to cobalt-rich portions of the interface. With the expected variation of atomic structure and the existence of a phase transformation at this M-S interface (both discussed earlier), the situation is expected to be quite complicated. The SBHs of single crystal CoSi, layers on Si(1 lO)andSi(lOO)aresimilar,both-0.7 eV on an n-type substrate. 8.5.5.2 Epitaxial Elemental Metals

The epitaxial interfaces Al/Si (1 1 1) and Ag/Si (1 1 1) have been studied by HREM (LeGoues et al., 1986,1987). The high density of interface steps and defects has so far prevented a conclusive understanding of the interface structures. Al/Si( 111) interfaces formed by a partially ionized beam appear to be flatter, but incommensurate (Lu et al., 1989). Depending on the growth conditions, an AI-Ga exchange reaction takes place at the AVGaAs interface (Landgren and Ludeke, 1981). There is also a noted reaction at the interface of Fe/GaAs (Krebs et al., 1987). The interfaces between epitaxial aluminum and GaAs (100) have been studied by a number of techniques (Marra et al., 1979; Kiely and Cherns, 1988). An MBE prepared GaAs ( 100)surface may have a variety of reconstructions which are associated with different surface stoichiometries. The periodicities of some of these superstructures are found to be preserved at the Al/GaAs interface (Mizuki et al., 1988). It has been discovered that the Schottky barrier height between epitaxial aluminum of silver layers and GaAs is a function of the original GaAs surface reconstruction/stoichiometry (Cho and Dernier, 1978; Ludeke et al., 1982; Wang, 1983). However, different conclusions have been drawn in other studies (Barret and Massies, 1983; Svensson et al., 1983; Missous et al., 1986).

524

8 Interfaces

8.5.5.3 Intermetallic Compounds on III-V Semiconductors The interfaces between GaAs (AlAs) and a number of epitaxial intermetallic compounds have been examined by HREM (Harbison et al.. 1988; Palmstrom et al., 1989a, b: Sands et al., 1990; Tabatabeie et al., 1988; Zhu et al., 1989). However, the atomic structures of these interfaces have not been properly modeled. Evidence for a reconstruction has been observed at the interface between NiAl and overgrown GaAs (Sands et al., 1990). Negative differential resistance has been observed at a discrete bias in electrical transport perpendicular to the interfaces of a GaAs/AlAs/NiAl/ AlAs/GaAs structure (Tabatabaie et al., 1988). This is thought to indicate quantization of states in the thin NiAl layer (Tabatabaie et al., 1988). 8.5.6 Conclusions

We have briefly reviewed important developments in the field of epitaxial metalsemiconductor structures. Obviously, much has been accomplished in the fabrication

and characterization of these structures. But the need still exists for better structures and better electrical properties. Further studies should prove to be beneficial both to our understanding of the SBH mechanisms and, perhaps, to next generation devices.

8.6 Interfaces Between Crystalline and Amorphous Materials: Dielectrics on Silicon 8.6.1 The Si/Si02 System The Si/Si02 interface, which forms the heart of the gate structure in a metal-oxide - semiconductor field effect transistor (MOSFET), is arguably the most economically and technologically important interface i n the world. The MOSFET, depicted in Fig. 8-5 1 , has enabled the microelectronics revolution, and unique attributes of the Si/Si02 interface, such as ease of fabrication and low interface state density, have made this possible. The dimensions of MOSFETs and other devices have continuously shrunk since the advent of integrated circuits about forty years ago, according to

Figure 8-51. Schematic diagram of a simple n-channel MOSFET.

8.6 Interfaces Between Crystalline and Amorphous Materials: Dielectrics on Silicon

525

Table 8-3. Technology roadmap characteristics in the area of thermavthin films. First year of production DRAM generation Minimum feature size (nm) Equivalent oxide thickness (nm)

1997

1999

2001

2003

2006

2009

2012

256M 250 4-5

1G 180 3-4

1G 150 2-3

4G 130 2-3

16G 100 1.5-2

64G 70 8) dielectric constant, (b) thermodynamic stability with respect to the formation of S i 0 2 , silicides, or oxides of other stoichiometry (i.e., they should not react with the underlying silicon), (c) a low concentration of interface states, (d) resistance to dopant and other impurity diffusion, (e) low leakage current, and (f) a large band gap (> 4 eV) and a large conduction band offset relative to silicon.

8.7 Conclusion In this chapter, we have attempted to outline the major structural characteristics of four generic classes of interface systems. For lattice-matched systems, high resolution electron microscopy techniques have been developed to probe the chemical structure across heterojunctions on the atomic scale. Recent electron holography results have demonstrated the potential to characterize and understand field distributions at interfaces defined by different dopants. Future evolutions of these techniques will enable greater understanding of the correlation between structural and electronic and optical properties of these systems. For latticemismatched isostructural systems, detailed mechanistic understanding of the primary relaxation modes, roughening, and misfit

14

21

-30

25

I

14

dislocation injection, is evolving. Further, ever-increasing capabilities in atomistic simulations of defects, coupled with experiment, are enabling the understanding of the electronic and optical properties of interfacial dislocations to be refined. An enduring challenge, however, is the ability to control dislocation densities in high mismatch heterostructures. For these systems, a “floor” density of the order of lo6 cm-, of threading defects is observed. For heterostructures consisting of dissimilar crystalline materials, such as epitaxial metal silicides on silicon, a major issue is the difficulties associated with wetting in many systems with high interfacial energy. Understanding and control of systems with multiple epitaxial relationships is also a major challenge. The structural quality attainable in some systems, such as NiSi,/Si and CoSi,/Si, however, has enabled significant advantages in the understanding of the fundamental mechanisms of Schottky barrier formation. Finally, the archetypal interface between crystalline and amorphous materials is the Si02/Si interface, which is of critical importance to the entire microelectronics industry. Although this is undoubtedly one of the most extensively studied materials systems in existence, much needs to be learnt about this interface and the fundamental mechanisms of silicon oxidation. The need to maintain ultrahigh perfection of electronic and dielectric properties as gate oxides in transistors become ever thinner is driving the industry to explore and apply new gate dielectric materials such as silicon oxynitrides.

8.8 References

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9 Material Properties of Hydrogenated Amorphous Silicon

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.

R A Street and K Winer Xerox Palo Alto Research Center. Palo Alto. CA. U.S.A.

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 9.1 9.1.1 Plasma-Enhanced Chemical Vapor Deposition Growth of Hydrogenated Amorphous Silicon . . . . . . . . . . . . . . . . . . . . 546 9.1.2 Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 550 9.1.3 Chemical Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Localization of Electronic States . . . . . . . . . . . . . . . . . . . . . . . 552 Electronic Structure and Localized States . . . . . . . . . . . . . . . . . 553 9.2 553 9.2.1 Band Tail States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 9.2.2 Doping and Dopant States . . . . . . . . . . . . . . . . . . . . . . . . . . 555 9.2.2.1 The Doping Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2.2 Dopant States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 558 9.2.3 Native Defects and Defect States . . . . . . . . . . . . . . . . . . . . . . 9.2.3.1 Microscopic Character of Defects . . . . . . . . . . . . . . . . . . . . . . 558 9.2.3.2 Dependence of the Defect Concentration on Doping . . . . . . . . . . . . 558 9.2.3.3 Distribution of Defect States . . . . . . . . . . . . . . . . . . . . . . . . . 559 9.2.3.4 Dependence of the Defect Concentration on Growth Conditions . . . . . . 561 9.2.4 Surfaces and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 561 9.2.4.1 Surface States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 9.2.4.2 Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4.3 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 563 9.2.5 Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 9.3 Conductivity, Thermopower and Hall Effect . . . . . . . . . . . . . . . . . 564 9.3.1 566 9.3.2 The Drift Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defect Equilibrium and Metastability . . . . . . . . . . . . . . . . . . . 568 9.4 9.4.1 The Hydrogen Glass Model . . . . . . . . . . . . . . . . . . . . . . . . . 568 9.4.2 Thermal Equilibration of Electronic States . . . . . . . . . . . . . . . . . 570 9.4.3 The Defect Compensation Model of Doping . . . . . . . . . . . . . . . . . 571 9.4.4 The Weak Bond Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 9.4.4.1 The Distribution of Gap States . . . . . . . . . . . . . . . . . . . . . . . . 575 9.4.5 Defect Reaction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 576 9.4.5.1 Stretched Exponential Decay . . . . . . . . . . . . . . . . . . . . . . . . . 577 9.4.5.2 Hydrogen Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Metastability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 580 9.4.6.1 Defect Creation by Illumination . . . . . . . . . . . . . . . . . . . . . . .

542 9.4.6.2 9.5 9.5.1 9.5.2 9.5.2.1 9.5.3 9.6 9.7


Defect Creation by Bias and Current . . . . . . . . . . . . . . . . . . . . . Devices and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . Thin Film Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . P-i-n Photodiodes and Solar Cells . . . . . . . . . . . . . . . . . . . . . . The Photodiode Electrical Characteristics . . . . . . . . . . . . . . . . . Matrix Addressed Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

582 583 583 584 . 585 589 592 593


List of Symbols and Abbreviations average recombination constant distance between sites creation probability of defects sample thickness characteristic distribution coefficient negatively charged defect states prefactor of diffusion coefficient hydrogen diffusion coefficient phosphorus distribution coefficient electron charge energy from the mobility edge energy activation energy of conductivity activation energy barrier of a thermal defect creation process mobility edge energy of conduction band slope of conduction band tail slope of valence band tail gap state energy associated with UDo demarcation energy Fermi energy energy to release hydrogen from a Si-H bond gap state energy associated with donor trap depth below conduction band mobility edge energy of valence band gap state energy of the valence band tail state energy provided by recombinations conductivity activation energy shift of defect band shift in Fermi energy change of energy of the ion core interaction with and without a defect Fermi function electric field Land6-g-factor illumination intensity hyperfine splitting nuclear spin crystal momentum Boltzmann constant time dependent rate constant reaction constant charge carrier density carrier density prefactor number of valence electrons

543

544


density of states distribution function distribution of conduction band energies distribution of trap energies concentration of the 4-fold silicon sites distribution function of the formation energy density of excess electrons in band edge states effective conduction band density of states defect concentration time dependent defect density decrease of excess electron density neutral defect density negatively charged defect density equilibrium defect density neutral defect density for unshifted Fermi energy dopant concentration hydrogen concentration trap concentration band tail density-of-states prefactor hole concentration Si-Si bond length cooling rate time temperature effective temperature of trap distribution slope of the exponential conduction band tail freezing temperature glass transition temperature parameter describing the Meyer-Neldel rule slope of the exponential valence band tail defect correlation energy formation energy of the defect formation energy of the neutral defects formation energy of the donor formation energy of the neutral dopant volume disorder potential glass volume lattice relaxation energy r.f. plasma power gas phase mole fraction mole fraction of impurity I mole fraction of P coordination number, number of neighbors in amorphous material


dispersion parameter, T/To deposition rate per unit r.f. plasma power pyrolytic rate constant stretched exponential parameter, T/T, temperature coefficient of Fermi energy bond angle between neighboring Si-atoms doping efficiency, [P,]/[P] dihedral angle charge carrier mobility free mobility of charge carriers effective drift mobility of charge carriers mobility prefactor width of gaussian distribution of defect states conductivity conductivity prefactor minimum metallic conductivity conductivity constant decay time free carrier lifetime lifetime of carrier in traps relaxation time rate prefactor, attempt-to-escape frequency 5 ( T , T,, NH) entropy factor a-Ge :H a-Si a-Si :H a-Sic :H a-SiGe :H a-SO,: H a-SiN,: H c-Si d.c. ESR MBE PECVD RDF r.f. r.m.s. TFT U.V.

"PPm

hydrogenated amorphous germanium amorphous silicon hydrogenated amorphous silicon amorphous silicon carbide amorphous silicon-germanium alloy amorphous silicon oxide amorphous silicon nitride crystalline silicon direct current electron spin resonance molecular beam epitaxy plasma-enhanced chemical vapor deposition radial distribution function radio frequency root mean square thin film transistors ultra violet parts per million of volume

545

546


9.1 Introduction Hydrogenated amorphous silicon (a-Si:H) has been actively studied for about 30 years. The unhydrogenated material (a-Si) has such a large defect density that it is unusable for electronic devices, although there is continuing interest in its atomic structure. The beneficial effects of hydrogen were discovered at least in part by accident, when the material was deposited from silane (SiH,) gas in aplasma discharge. This method of growth results in a reduction of the defect density by about four orders of magnitude compared to unhydrogenated a-Si, giving material which is of device quality. It is now recognized that hydrogen removes defects by bonding to unterminated silicon atoms. The first demonstration of substitutional doping, made possible by the low defect density, was reported in 1975 and opened the way to many device applications (Spear and LeComber, 1975). Since that time the research effort has greatly expanded so that this material now dominates studies of amorphous semiconductors, and many technology applications have developed. The first photovoltaic solar cells made from a-Si : H were described by Carlson and Wronski (1976), and the conversion efficiency has steadily improved to its present value of 13-14%. In 1979 the plasma deposition of silicon nitride was used in conjunction with a-Si : H to produce field effect transistors (LeComber et al., 1979). Large area arrays of these thin film transistors (TFI') are now important i n liquid crystal displays (Miki et al., 1987) and monolithic circuits for printing and input scanning applications (Thompson and Tuan, 1986). The combination of doping and transistor action means that essentially all of the circuit elements used in crystalline silicon electronics can be reproduced in a-Si : H,

giving it a broad versatility in electronic circuit design. The principal advantage of using a-Si : H is that it can be deposited over large areas on low cost substrates such as glass.

9.1.1 Plasma-Enhanced Chemical Vapor Deposition Growth of Hydrogenated Amorphous Silicon The usual method of depositing a-Si :H is by plasma decomposition of silane gas, SiH,, with other gases, such as PH,, B,H6, GeH, etc., added for doping and alloying. Silane decomposes thermally above about 450°C and amorphous films can be grown in this way at temperatures less than about 550°C. However, these films are of limited utility because the temperature is too high to retain the hydrogen. The deposition of hydrogenated films at lower temperatures requires a source of energy to dissociate the SiH,, which is the role of the plasma. The first plasma deposition system for amorphous silicon was a radio frequency (r.f.) inductive system developed by Chittick et al. (1969). Most subsequent reactors are in a diode configuration in which the plasma is confined between two parallel electrodes. This type of reactor is illustrated in Fig. 9- 1, and consists of a gas inlet arrangeHeater

r Gas inlet - f l

I Pump

Figure 9-1. Schematic diagram of the main components of a typical r.f. diode plasma reactor for depositing a-Si : H and its alloys.

547

9.1 Introduction

ment, the deposition chamber which holds the heated substrate, a pumping system, and the source of power for the discharge. The deposition process is usually referred to as plasma-enhanced chemical vapour deposition (PECVD). There are numerous variations on this basic deposition process, but the resulting material is largely independent of the choice of technique. The structural and electronic properties of the film depend on the conditions of growth, particularly substrate temperature, r.f. power and gas composition. Figure 9-2 summarizes some typical measurements of PECVD a-Si :H films (Knights and Lucovsky, 1980). The hydrogen content varies between 8 and 40 at.% and decreases slowly as the substrate temperature is raised. In addition, the hydrogen content depends on the r.f. power in the plasma and on the composition of the gas; Fig. 9-2b shows the

variation when SiH, is diluted to 5% concentration in argon. The defect density also depends on the substrate temperature and power with variations of more than a factor 1000. The minimum defect densities of 1015-1016cm-3 usually occur between 200 and 300°C and at low r.f. power densities (e100 mW/cm2), giving material with the most useful electronic properties. Typical growth rates range between 1 and 10 f \ / s and are approximately proportional to the r.f. power, provided that the SiH, is undiluted. Heavy dilution with argon and high r.f. power give depletion effects in which the SiH4 is completely consumed in the plasma and the deposition rate saturates (Le. does not increase with power). Good quality material is grown far away from this condition. The physical morphology of the films also depends on the deposition conditions.

+

2( s 4

0

0

.-C

.-C

.\" LC

C

C

z

0

z 15

30

E

I

L

C W

C

u

V

c

C 0

0

10

20

W C

0)

cn

m

-0

U >r I

2

P

x I

10

5

i 0

I

I

100

I

I

200

I

I

I

300

Temperature in

OC

I

LOO

C

A 1015 10

20

30

rf power in W

Figure 9-2. Illustration of the dependence of material properties on PECVD deposition conditions, showing variations of the hydrogen concentration and defect density on (a) substrate temperature, and (b) r.f. power (Knights and Lucovsky, 1980).

548


Low r.f. powers and undiluted SiH4result in smooth conformal film growth and material with a low defect density. In contrast, high r.f. power and SiH, diluted in argon give films with a strongly columnar microstructure oriented in the direction of growth and a high defect density (Knights and Lujan, 1979). The columns have dimensions ranging from about 100 A to 1 pm. The smooth growth habit is important for a-Si : H-based devices and multilayer structures whose layer thicknesses are typically only a few hundred Angstrom. Film growth occurs because SiH, is dissociated by the plasma and the fragments condense out of the gas onto the substrate and other reactor surfaces. Although the principle of deposition is quite simple, the physical and chemical processes which take place during a-Si:H film growth are complex. A plasma is sustained by the acceleration of electrons due to the alternating electric field. The electrons collide with the molecules of the gas, causing ionization (amongst other processes) and releasing more electrons. There is negligible acceleration of the ions in the plasma by the electric field because of their large mass, so that the energy of the plasma is acquired by the electrons. In addition to ionization, the collisions of the energetic electrons with the gas molecules cause dissociation of the gas, creating either neutral radicals or ions which are the precursors to deposition. Examples of silane dissociation reactions which require low energies are SiHj + SiH2 + H2 SiHl

+ SiH, + H

(+2.2 eV)

(9- 1 )

(+4.0 eV)

(9-2)

In addition, many other higher energy silane dissociation and reaction processes have been identified. At the normal deposition pressures of 0.1-1 Torr, the mean free path of the gas

molecules is 10-3-10-2 cm, which is much smaller than the dimensions of the reactor. Many intermolecular collisions take place, therefore, in the process of diffusion to the substrate. An understanding of the growth is complicated by these secondary reactions because they greatly alter the mix of radicals. Those with a high reaction rate have a low concentration and short diffusion length and so are less likely to reach the growing surface. The least reactive species tend to survive the collisions longest, and have the highest concentrations, irrespective of the initial formation rates. SiH, radicals with 1s 2 readily react with SiH, to form Si2H4+,, whereas SiH, does not react with SiH,, because Si,H, is not a stable structure. Thus, it is believed that SiH, is an important precursor to growth, particularly in low power plasmas (Gallagher, 1986). The growth of a-Si :H is completed when molecular fragments are adsorbed onto the growing surface, with the concomittant release of atoms or molecules from the surface. Figure 9-3 illustrates some of the processes which have been proposed to occur at the growing surface of an a-Si:H film (Gallagher, 1986). Of all the gas species near the surface, atomic hydrogen can penetrate furthest into the material. At the normal deposition temperature of 200-300°C, interstitial hydrogen can move quite rapidly into the bulk where it readily bonds to silicon dangling bonds. Hydrogen, therefore, has the fortunate property of being able to remove most of the subsurface defects left by the deposition process. The microstructure of the film reflects the growth process. Columnar growth is caused by a shadowing effect and occurs when the impinging molecular fragments have a very low surface mobility. Any depression in the surface is shadowed from the plasma by the surrounding material, which suppresses the growth rate and accentuates the depression.

9.1 Introduction

Non-bonding of SiH, to Hterminated

Bonding of SiH, to surface

The result is a columnar structure containing many voids and defects. The shadowing is prevented when the growth species have a high surface mobility. SiH3 is believed to have this property because it is less reactive than the other radicals.

9.1.2 Molecular Structure Bonding is primarily covalent in a-Si and the tetrahedral arrangement of bonds characteristic of crystalline Si (c-Si) is essentially preserved in the amorphous phase. The bulk electronic and vibrational densities of states are, to a good approximation, broadened versions of their crystalline counterparts. Disorder in a-Si is best characterized, therefore, in terms of deviations from the perfect diamond structure of c-Si. Diffraction measurements can provide only average measures of these deviations, which has led to the development of sophisticated model building techniques to uncover the microscopic structure of a-Si (Polk, 1971). A computer-generated picture of the result of such a model building exercise is shown in Fig. 9-4. This hand-built, 105-atom ball-and-stick model of a-Si: H with periodic boundaries permits the calculation of a-Si: H properties and provides a wealth of microscopic structural information unavailable by other means (Winer and Wooten, 1984). It is clear from Fig. 9-4 that aSi :H is both structurally and compositionally disordered. The hydrogen in a-Si :Hreduc-

549

Figure 9-3. Illustration of some possible processes taking place at the a-Si : H surface during growth (Gallagher, 1986).

es the network strain which allows the material to be doped both n- and p-type. Experimental information about the local order of silicon atoms comes from the radial distribution function (RDF) which is the average atomic density at a distance r from any atom. The RDF of a-Si:H shown in Fig. 9-5 (Schulke, 1981), has sharp structure at small interatomic distances, progressively less well-defined peaks at larger distances, and is featureless beyond about 10 A. This reflects the common property of all covalent amorphous semiconductors, that the short-range order of the crystal is preserved but the long-range order is lost. The Si-Si bond length, R,,given by the first peak in the RDF is the same as in crystalline silicon, 2.35 A, and the intensity of

Figure 9-4. Computer generated picture of a handbuilt, 105 atom model of a-Si :H with periodic boundaries.

550


12 ?

7 0.E P E 0.1 U

a 0

0

Distance in

A

Figure 9-5. Example of the radial distribution function (RDF) of a-Si : H obtained from X-ray scattering. The atomic spacings which correspond to the RDF peaks are indicated (Schulke, 1981).The arrows indicate the range of values of the third nearest neighbor as the dihedral angle 4 is varied.

the first peak confirms the expected 4-fold coordination of the atoms. The bond length distribution is evidently small and is estimated to be about 1% (Etherington et al., 1984). The second peak in the RDF arises from second neighbor atoms at a distance 2R, sin(8/2), where 8 is the bond angle. The second peak occurs at 3.85 A, which is also the same as in crystalline silicon, giving an average bond angle of 109O, and establishing the tetrahedral bonding of a-Si : H. The width of the second peak indicates that the root mean square (r.m.s.) bond angle deviation is about 10". The third neighbor peak in the RDF is even broader. The distance depends on the dihedral angle, cp (see Fig. 9-5) and a very broad distribution of angles about the crystalline silicon values is deduced from the position and width of the peak. The third neighbor peak and the more distant shells overlap, so that no detailed information can be deduced. An idealized, though useful, way to view a-Si:H is as a homogeneous collection of Si-Si and Si-H bonds rather than as a collection of individual atoms. The non-equilibrium nature of the typical a-Si : H growth process prevents the structure from attain-

ing its thermodynamic ground state (i.e. the crystal) resulting in a large (= 0.2 eV/atom) excess enthalpy of disorder. Under typical growth conditions, large-scale heterogeneities can also form such as voids and cracks. Nevertheless, the structure of a-Si :H is remarkably close to the diamond structure with most bond lengths within 1 % and most bond angles within 10%of their c-Si values. These relatively small strains are responsible for the localization of gap states that are the distinguishing feature of the electronic structure of a-Si : H. 9.1.3 Chemical Bonding A molecular orbital representation of the conduction and valence bands of a-Si : H is illustrated in Fig. 9-6. These s and p states combine to form the sp3 hybrid orbitals characteristic of ideal tetrahedral bonding (see Chap. 1). These orbitals are split by the bonding interactions to form the valence and conduction bands. Any non-bonding silicon orbitals, such as dangling bonds, are not split by the bonding interaction and give states in the band gap. The model in Fig. 9-6 is sufficient to predict the general features of the density-of-states distribution, but much more detailed calculations are needed to obtain an accurate distribution. Many calculations have been performed (Allen and Joannopoulos, 1984), but present theories are not yet as accurate as the corresponding results for the crystalline band structure. The Si-H bonds give further contributions to the electronic structure, with bonding states deep in the valence band and antibonding states near the conduction band edge (Ching et al., 1980). The atomic structure of a covalent amorphous material is described by a continuous random network as illustrated in Fig. 9-4, in which each atom is bonded to a welldefined number of neighbors - the coordi-

9.1 Introduction I I I

;

’ A m

!

,’ onti-

-.P

A \\\ .

..

, 4

S

-0

Figure 9-6. Schematic molecular orbital model of the electronic structure of amorphous silicon

bonding

1 ,I

t

conduction bonds

551

’

, ,,

Atom

mobility edge

’,

bonding

k\F Solid

valence r

Density of states)

nation number, Z. In the absence of the topological constraints of the crystalline lattice, the local bonding of each atom is determined by the optimum chemical configuration. A covalently-bonded atom has minimum energy when the maximum number of electrons are in bonding states. The most favorable coordination of atoms from group IV to VI1 of the periodic table is Z=8-N

(9-3)

where N is the number of valence electrons ( N = 4 for Si etc.). This “8-N” rule is a useful description of the bonding chemistry of amorphous networks containing multiple elements (Mott, 1969). The effects of the bonding chemistry in amorphous and crystalline materials are different, due to the absence of a periodic structure. This is illustrated by considering impurity atoms. Since every atomic site in a crystal is defined by the periodic lattice the impurity either substitutes for the host adapting itself to the chemistry of the host or occupies a position which is not a lattice site, forming a defect. A substitutional impurity such as phosphorus is 4-fold coordinated in crystalline silicon and acts as a donor by releasing one of its electrons into the conduction band. An amorphous material has no rigidly defined array of lattice

sites, so that an impurity can adapt the local environment to optimize its own bonding configuration, while also remaining a part of the host atomic network. The 8-N rule predicts that phosphorus in amorphous silicon should be 3-fold coordinated and, therefore, inactive as an electronic dopant. Indeed, it seems to follow from the 8-N rule that substitutional doping must be impossible in an amorphous semiconductor. Actually, the chemical bonding does not forbid, but does severely constrain doping in a-Si :H, as discussed in Sec. 9.4.2. The possibility of electronically-induced structural reactions is also suggested by the 8-N rule, which predicts that the excitation of an electron out of a state can destabilize the chemical bond and, thereby, induce a change in coordination. Such reactions are usually prevented in crystalline semiconductors by the long-range order of the lattice and the extended electronic wavefunctions. Structural changes are promoted in amorphous materials by the adaptibility of the continuous random network and by the localization of electronic carriers. The various metastable and equilibrium processes described in Sec. 9.4 follow from this property.

552


9.1.4 Localization of Electronic States The periodic potential in a crystal leads to the familiar Bloch wavefunctions which are extended states defined by the crystal momentum, k . These solutions to Schrodinger's equation do not apply to an amorphous semiconductor because the potential is not periodic. The structural disorder of an amorphous semiconductor causes such frequent electron scattering that the wavefunction loses phase coherence over a distance of a few atomic spacings. The uncertainty in the momentum arising from the scattering is of order k , so that momentum is not a good quantum number and is not conserved in electronic transitions. Consequently the energy bands are no longer described by the energy-momentum dispersion relations, but instead by a density-of-states distribution, N ( E ) , illustrated in Fig. 9-6, and the symmetry selection rules do not apply to optical transitions. The disorder also reduces the carrier mobility and causes localization of the electron wavefunctions. Anderson (1958) showed that all the electronic states are localized when there is a large enough site-to-site variation in the atomic potentials, V,,. The condition for complete localization is that V , exceeds about three times the electronic band width. The disorder in the amorphous semiconductors is not sufficient to localize all the electron states, primarily because the short range order limits the magnitude of the disorder potential. Instead, the center of the band comprises extended electron states at which there is strong scattering, while states at the extreme edges of the bands are localized (Mott and Davis, 1979; Chap. 1). The density-of-states distribution in the region of the band edges is shown i n Fig. 9-7. The extended and localized states are separated by a mobility edge, at energy E, and E,, which derives its name because at zero

10'1

'

OL

'

02

'

0

'

\,

02

"

I

04 16 Energy in eV

l

l

1

18

20

22

Figure 9-7. Density of states distribution near the band edges, showing localized band tail states extending into the band gap of a-Si :H. Eov and E,, are the slopes of the exponential band tails; E, and Ev are the energies of the valence and conduction band mobility edges.

temperature, only electrons above E, are mobile and contribute to the conduction. The energy of the mobility edge within the band depends on the degree of disorder and is typically 0.1-0.5 eV from the band edge in all the amorphous semiconductors. The detailed properties of states near the mobility edge are much more complicated than in this simple model of an abrupt mobility edge (Abrahams et al., 1979; Mott and Kaveh, 1985). However, the simple model is sufficient to interpret most experimental measurements. The reason that electronic conduction in a-Si : H is not metallic is therefore that electronic states in the gap are localized and conduction of charged carriers through these states occurs at elevated temperatures via thermal excitation to the mobility edges or by hopping from site to site. As the conducting states become less localized (toward the band edges) and the site separation decreases, the carrier mobility increases until extended state conduction is achieved. The region between E , and E, in Fig. 9-7

0.2 Electronic Structure and Localized States

is referred to as the “mobility gap”, and is of magnitude 1.8- 1.9 eV (Wronski et al., 1989).


553

ization of electronic states near the valence and conduction band edges as well as states associated with defects and impurities. The densities and distributions of localized states within the fundamental band gap aSi : Hare of primary importance in determining the optical and electronic limitations of any a-Si :H-based technology.

The distribution of electronic states in a-Si:H is similar to that of crystalline Si except that critical point structure is absent in the amorphous phase due to the lack of translational symmetry and a well-defined band structure. Similarly, transitions between electronic states in a-Si:H are not subject to symmetry selection rules or crystal momentum conservation requirements, which leads to the direct nature of band gap transitions in a-Si : H and higher absorption coefficients than c-Si in the visible. The most important distinction between the electronic structure of a-Si:H and that of c-Si, however, is the localization of a small but important fraction (~0.02%)of the electronic states in a-Si :H. The distributions of bond lengths and bond angles which characterize a-Si : H disorder lead to the local-

The distribution of band tail states is most easily, although indirectly, observed in the optical absorption spectrum, which in the case of a-Si :H and a host of other glassy and crystalline materials exhibits characteristic exponential absorption of Urbach edge as shown in Fig. 9-8. Several explanations of the exponential nature of the Urbach edge have been proposed many relying on strong electron-phonon or excitonic mechanisms of band edge broadening (Mott and Davis, 1979; Sec. 9.6). While such explanations apply to the case of ionic crystalline materials such as the alkali halides, it is now well established that the densities of electronic states themselves tail exponentially into the

Photon energy in eV

Figure 9-8. The optical absorption edges of various amorphous semiconductors including a-Si : H (Mott and Davies, 1979).

9.2.1 Band Tail States

554


band gap in a-Si : H as indicated in Fig. 9-7. The conduction band tail decreases exponentially into the gap with a characteristic inverse slope E,, between 20 and 30 meV, or equivalently, with a characteristic temperature T, (E,,=kT,) near 300 K. The valence band slope is generally broader with a characteristic inverse slope Eov between 40 and 50 meV, i n the best material, corresponding to Tv near 500 K. Lower quality material has EOc up to 150 meV. For a long time it was not clear why the disordered nature of a-Si should lead to a gap in the density of states. Weaire (1971) used a simple two-parameter tight-binding Hamiltonian to model a-Si and showed that for reasonable levels of off-diagonal disorder forbidden energies always appear in the density-of-states. In real a-Si : H the valence and conduction band tails intersect at some very low density-of-states and the Fermi energy lies deep in the gap. The band tails are the most prominent feature in the mobility gap and have a dominant effect on important processes such as optical absorption, electron-hole recombination, and electronic transport in a-Si : H. Their exponential distribution arises from the distribution of weak (strained) Si-Si bonds whose bonding and anti-bonding states make up the valence and conduction band tails, respectively. If a Si-Si bond becomes too strained, it can break to reduce the system free energy and, thereby, form a network defect whose state is localized deep in the mobility gap. The electronic states of substitutional impurities are also localized similar to the shallow donor and acceptor states in c-Si. In fact. the behavior of dopants and defects in a-Si : H is qualitatively the same as that in c-Si, but the uniform disorder of a-Si : H leads to several interesting differences that have only recently become understood, and which are discussed in Sec. 9.4.

9.2.2. Doping and Dopant States

A-Si : H, like c-Si, can be doped either nor p-type by the incorporation of phosphorus or boron, respectively, during growth or by ion-implantation after growth. The solubility of impurities in a-Si :H is higher than in c-Si and P or B concentrations up to lo2’ cm-3 are easily achieved. In c-Si, every P or B atom is incorporated at a substitutional lattice site which constrains all impurities to be 4-fold coordinated and, therefore, active donors or acceptors. In aSi : H, the network disorder provides a certain structural flexibility which allows a large fraction of the impurities to achieve their preferred valency; i.e. 3-fold coordination in the cases of P or B. For a long time this flexibility was thought to preclude the significant doping of amorphous semiconductors. However, the strain relief provided by hydrogenation permits doping efficiencies as high as 1 % in a-Si : H. The best evidence of doping is the increase of the d.c. conductivity of a-Si :H observed when phosphine or diborane is added to the silane plasma during film growth (Spear and LeComber, 1975). The room temperature dark d.c. conductivity Q-’ cm-’ in increases rapidly from = intrinsic a-Si :H up to = lo-* Q-’ cm-’ in highly P-doped a-Si:H as shown in Fig. 9-9. The exponential dependence of the conductivity on inverse temperature is characteristic of the thermally-activated charge transport and demonstrates that the Fermi energy lies within the mobility gap. The “kinks” in the a-Si :H conductivity curves and thermopower near 150°C are similar to those observed in the conductivity of crystalline silicon or even gold (Hannay, 1967). In these crystals, the conductivity is limited by carrier scattering from point defects, and “kinks” occur in the conductivity curves when the temperature drops be-


102

-

Figure 9-9. Temperature dependence of the conductivity of phosphorus- and borondoped a-Si : H. The doping levels range from to 2x lo-’ (Beyer and Overhof, 1984).

102

1

1 c

5

.E

r

b

c: 10-2

.-

.-c

b

b

10-2

10-2

10-L

10-6

10-6

0

1

2 3 4 103/T in K-’

5

0

low the point where the defect formation reactions can no longer maintain equilibrium and non-equilibrium point defect concentrations are frozen. In the case of a-Si :H, the “kinks” occur at temperatures below which bulk dopant activation reactions, such as

P:

* P+:

e-

555

(9-4)

can no longer reach equilibrium and nonequilibrium carrier concentrations are frozen. The equilibrium concentration of charge carriers and the equilibration kinetics of such reactions are discussed in detail in Sec. 9.4. 9.2.2.1 The Doping Efficiency

The fraction of bulk impurities which are active dopants, termed the doping efficiency, is shown for arsenic-doped a-Si: H in Fig. 9-10. These results are derived from measurements of the concentration of excess electrons introduced by the doping. We might expect that only a limited number of the sites are available at which impurities in a-Si : H can be constrained to 4-fold coordination, so that the doping efficiency should increase with decreasing impurity concen-

1

2 3 L 103/T in K-’

5

tration. This is exactly what is observed under most plasma-enhanced deposition conditions, where the doping efficiency decreases as the square-root of the gasphase mole fraction X , of impurity, I (=XI) (Stutzmann et al., 1987). However, at very low a-Si:H growth rates (low r.f. power), another, pyrolytic process of inactive impurity incorporation acts to reduce the doping efficiency at low XI. This behavior can be understood with the following analysis. The doping efficiency of P, for example, in a-Si:H is defined by q = [P,]/[P], where the square brackets denote the solid phase atomic density. In order to interpret the data in Fig. 9-10, we want to rewrite q in terms of the gas-phase P mole fraction X,. The P distribution coefficient d(P) E [P]/[Si] X, relates the solidphase P density [PI to X,. A simple twoparameter analytical expression for d (P) has been fitted to a large set of data to obtain (Winer and Street, 1989) (9-5) where do is a constant which is characteristic of the relative plasma-enhanced dissoci-

556


As-doped a-Si : H

Figure 9-10. Doping efficiency of arsenic in a-Si : H as a function of the mole fraction of As in the gas during deposition (Winer and Street, 19891, for different r.f. powers.

Gas-phase mole fraction of As

ation rates of phosphine and silane, p is the pyrolytic rate constant, and a is the deposition rate per unit r.f. plasma power WrF This relation describes the competition between impurity incorporation via pyrolysis (= p Xi”*)and via plasma-enhanced decomposition (= a W). When the latter dominates (at high r.f. plasma power and large X,), the distribution coefficient is essentially constant between 5 and 10. When thermal decomposition of the impurity gas (i.e., B2H6,ASH,, PH,, etc.) dominates impurity incorporation (low r.f. plasma power and small X ) , the distribution coefficient can attain values near 30 for P (psmall)and near 300 for As ( p large). Finally, active dopant incorporation can be described by the chemical reaction Pgas% P i + e-

(9-6)

so that by the law of mass action at equilibrium and [P,‘] = [e-] one obtains [Pi] = X,”’ as observed experimentally (Street, 1982). We can now express T,J completely in

terms of X, to obtain

The same competition between plasma-enhanced decomposition and pyrolysis determines the doping efficiency in a-Si :H. Under conditions where plasma decomposition dominates such a high growth rates or small p’s (Le. B or P doping) the doping efficiency is simply proportional to x-”*. Under conditions where thermal decomposition is significant such as low growth rates and large p’s (i.e. As doping), TJ, is constant and very small, as observed in Fig. 9-10.

9.2.2.2 Dopant States Four-fold coordinated dopants such as B, and P, form effective mass donors and acceptors with localized dopant states near the band edges, as in c-Si. The variations in


disorder potential broaden the dopant levels, similar to the band tailing of the silicon states. Donor states have been observed through their electron spin resonance (ESR) hyperfine interaction with the spin of donor electrons (Stutzmann and Street, 1985). Donor electrons in a-Si: H are more localized than in c-Si and the hyperfine interaction is correspondingly stronger. The donor band is deeper in the gap and broader than the shallow, well-defined donor levels in c-Si. Because the valence band is much broader than the conduction band tail in a-Si :H, hyperfine measurements have not been able to detect the corresponding B, donor states in B-doped a-Si :H. The probable distributions of P,, As, and B4 dopant states in highly-doped a-Si :H are shown in Fig. 9-1 1. The arsenic donor is deeper than the phosphorus donor because of differences in size, electronegativity and bond strength. The P! ESR hyperfine signal in a-Si:H exhibits two lines split by AH = 24.5 mT with a peak-to-peak line width of about 6 mT for each line (Stutzmann and Street, 1985). The two hyperfine lines are due to

1

,

,

,

iZV 0.1 0.2

,

.,

,

~

0.3 0 L 0.5 Energy in eV

/

,

,

4.3 -0.2 -0.1

, EC

Figure 9-11. Distributions of shallow donors states in doped a-Si : H. Such states are broadened relative to their crystalline silicon counterparts by the network disorder. Dashed lines represent the band tails (Street, 1987).

557

the interaction of the spins of electrons localized at P, sites with the spin I = 1/2 of the 31P nucleus. The 31P hyperfine signal increases with the gas-phase P mole fraction X,, but the concentration of neutral donors is much smaller than that of ionized donors, because of compensational defects. The distribution of electrons which occupy the conduction band tail states can be measured by photoemission yield spectroscopy (Winer et al., 1988). In undoped slightly P-doped a-Si : H, the expected Boltzmann fall-off in the density of occupied states above the Fermi energy is observed. However, as the gas-phase P mole-fraction increases beyond lo-,, the Fermi tail broadens considerably. Dividing the measured occupied density of states by the FermiDirac occupation function reveal an exponentially increasing total density of states above the Fermi energy consistent with the leading edge of the P, donor band of states. The density and location of donor states observed by this method agrees with those inferred from the ESR hyperfine measurements. The carriers near the band edge include those occupying dopant states as well as the intrinsic band tail states, and in n-type a-Si :H there are roughly equal concentrations of each, because the band tail overlaps the donor band. The density of shallow carriers is quite low because of the low doping efficiency and the compensation by defects. The doping effect in a-Si :H is consequently rather weak. For example, at a gas phase doping level of 1%, less than 1% of the phosphorus is in the form of 4-fold donor states, and of these donors, about 90% are compensated by deep defects. Of the remaining 10% of the donor electrons at the band edge, most occupy band tail states, so that the free electron concentration is only 1O4 of the impurity Concentration. The ]OW mobility Of free carriers in a-Si :H compared to c-Si, results

558


in the conductivity of n-type a-Si : H being more than five orders of magnitude below that of c-Si.

9.2.3 Native Defects and Defect States Unsatisfied or “dangling” bonds are the dominant native defect in a-Si : H: the amphoteric nature of this defect gives rise to either unoccupied (positively-charged), singly-occupied (neutral), or doubly-occupied (negatively-charged) electronic states. These localized states lie near the middle of the mobility gap and greatly affect the movement of the Fermi energy and transport and recombination processes. An understanding of the character and distribution of defect states and their dependence on growth conditions and doping level is essential for the successful application of a-Si : H and its alloys. Not surprisingly, a multitude of experimental methods have been developed and intensive effort has been expended in the pursuit of this understanding.

9.2.3.1 Microscopic Character of Defects Defect states are generally the most localized of all electronic states in a-Si : H, lying deep in the mobility gap at low densities. Most of our knowledge concerning the microscopic character of defects is derived from electron spin resonance (ESR) measurements. ESR measures both the concentration and the local environment of the neutral paramagnetic defects in a-Si:H. The concentration of charged defects can also be determined by ESR by illuminating the material with band gap light during the measurement to depopulate charged defects and make them spin active (Street and Biegelsen, 1982). The dominant defect in undoped a-Si : H has a nearly symmetric spin signal at a g-value of 2.0055 and a peak-to-

peak width of ~ 0 . 7 5mT. These characteristics are similar to those of the ESR signal from unterminated bonds at the c-Si/Si02 interface, which has led to the identification of the g=2.0055 signal in a-Si:H as due to undercoordinated or “dangling” bonds. It has also been proposed that native defects are better described by overcoordinated or “floating” bonds (Pantelides, 1986). Such floating bonds have many of the same properties as dangling bond except that they are predicted to be very mobile whereas dangling bonds should be essentially immobile. Recent calculations and defect diffusion measurements have all but ruled out the floating bond hypothesis (Fedders and Carlssen, 1989; Jackson et al., 1990); the dangling bond remains the only defect model that can consistently account for the properties of native a-Si :H defects.

9.2.3.2 Dependence of the Defect Concentration on Doping The incorporation of substitutional impurities during growth leads to an increased charge carrier concentration in both c-Si and a-Si : H. In a-Si : H, the majority of extrinsic charge carriers are taken up to form compensating charged defects. This process can be described schematically in the case of P doping by the following chemical reactions P,

Si,

P;t+ e-

(9-8)

+ e- +s Si,

(9-9)

%

where the subscripts refers to the coordination. The reactions are independent but they are linked by their dependence on the carrier concentration. The required coordination changes are accomplished via hydrogen bond diffusion, and the kinetics of this and similar reactions is determined by the kinetics of dispersive hydrogen diffusion. Applying the law of mass action to reactions (9-8)


559

pendences on growth conditions and doping determined by different laboratories usually agree. However, the number of different methods for measuring defect state distributions is nearly as large as the number of different laboratories interested in this problem! The defect distributions extracted or inferred from these many methods do not always agree and at least two distinct a-Si:H I schools have developed whose disagreea-Ge:H ,, I ment in the placement of defect state energies in the gap amounts to several tenths of 1051 ' ' ' ' J' U io-' 10-~ 10-~10-5 10-3 10-1 an electron volt (Lang et al., 1982, Okushi Bfi6 PH,.ASH, et al., 1982). The results of three typical Dopant gas concentration in c d experimental methods which provide a more or less consistent picture of the defect Figure 9-12. Defect concentrations as a function of gas phase mole fraction of boron, phosphorus, and arstate distribution in a-Si:H are shown in senic (Stutzmann et al., 1987). Fig. 9-13. The luminescence transition observed in P-doped a-Si :H at 0.8-0.9 eV (Fig. 9- 13) and (9-9) leads to the square-root depenhas been interpreted as a band-tail-to-defect dence of the defect [Si;] and active dopant transition (Street, 1982). The 0.9 eV energy, therefore, is a measure of the separation [Pi] concentrations on the solid-phase P concentration [PI, which is proportional to between the charged defect band peak and, the gas-phase phosphorus mole fraction most probably, the conduction band tail. The under most deposition conditions. This shape and the 0.3 eV width of the luminessquare-root dependence of [Si;] on XI is cence band should be characteristic of the shape of the defect band. The defect distriobserved for all substitutional dopants, as shown in Fig. 9-12. However, the high bution inferred from deep-level transient spectroscopy (Lang et al., 1982) and optical intrinsic defect density in a-Ge :H masks the doping dependence in this material. absorption measurements is consistent with the luminescence measurements: negatively-charged D- defect states lie in a broad 9.2.3.3 Distribution of Defect States band 0.8-1 eV below the conduction band mobility edge in P-doped a-Si :H. The optiThe defect concentration in undoped cal absorption data indicate a defect level at a-Si:H determined by ESR or optical absorption typically lies between 1015 and about the same energy. Similar measure10l6 cm-3 for optimized deposition condiments on p-type material find a defect levtions. In heavily-doped a-Si : H, the defect el which is well separated from the valence concentrations can increase to IO'* ~ m - ~ . band edge (Kocka et al., 1987). In summary, the majority of doping-induced states and The detection of such small defect concenthe Fermi energy lie opposite each other in trations by ESR and optical absorption measurements (Jackson and Amer, 1982) is the gap; the Fermi energy of doped a-Si : H relatively simple and has become so stanlies in a relative minimum in the gap state distribution. This picture has been condardized that the defect concentration de-

-

-

9 Material Properties of Hydrogenated Amorphous Silicon I

I

I

I

I

I

I Yl

-

CAPACITANCE

/

ABSORPTION

?.?E EXPERIMENT

-CALCULATION

1

1

1

1

I

I

I 1.8

2.0

TRANSIENT CAPACITANCE

h' I

I 0.6

0.8

1.0

I I 1.2 1.4 1.6 ENERGY eV

I

Figure 9-13. Typical data from which a consistent distribution of gap states has been inferred; obtained from luminescence, deep level transient spectroscopy, and optical absorption experiments (Street et al., 1985).E, is the defect level measured from E , and W is the contribution to the optical transition energies from lattice relaxation. The calculated absorption is a convolution of the conduction band and the defect level distribution.

undoped

firmed by several other types of capacitance and absorption measurements, as well as by photoemission spectroscopy measurements (Winer et al., 1988). A schematic description of the distribution of gap states in ptype, n-type, and intrinsic a-Si : H is shown in Fig. 9-14. The 0.1-0.3 eV difference between the negatively-charged (doubly-occupied) and neutral (singly-occupied) defect state energies determined from optical absorption data is a measure of the defect correlation energy, Uc, defined as the energy cost to plase a second electron on a singly-occupied, localized defect level (Jackson, 1982). Electron spin resonance and optical absorption measurements are also consistent with a small, positive value of U , (Street and Biegelsen, 1982; Dersch et al., 1980; Jackson, 1982). The doubly occupied defect gap state energy level lies above the singly occupied level by an energy U,, as depicted in Fig. 9-14. This view of the distribution of defect states in intrinsic a-Si:H is generally accepted. However, the defect energy levels observed in doped a-Si: H are not consistent with these observations in undoped material, and suggest instead a defect correlation energy that is negative. The apparently conflicting requirements of a small, positive correlation energy in intrinsic a-Si :H and the reversal of the defect energy order-

n-type

I

E,

Figure 9-14. Schematic distribution of defect state in the gap for n-type, intrinsic and p-type a-Si : H. The two levels of the dagling bond are separated by the correlation energy U,. The shift, AE, of the defect levels with doping given by Eq. (10-48) and dicussed in the text.

b>


ing in a doped a-Si:H can be resolved by invoking a broadly-distributed pool of virtual defect states within a chemical equilibrium framework of defect formation from which the system can choose to form defects in order to minimize the system free energy (Winer, 1989). This defect formation framework is discussed in detail in Sec. 9.4.

561

increases beyond the optimal range, increased desorption at strain-relieving species like hydrogen again lead to an increased defect concentration. More detailed models which attempt to account for the behavior of Fig. 9-2a as discussed elsewhere (Tanaka and Matsuda 1987; Winer, 1990).

9.2.4 Surfaces and Interfaces 9.2.3.4 Dependence of the Defect Concentration on Growth Conditions Thin films of a-Si :H are typically deposited onto heated (250°C) glass substrates by the r.f. plasma decomposition of silane at gas pressures near 100 mTorr. Under these conditions, growth rates of about 1 P\ per second are typically achieved. Ion bombardment, U.V. exposure, and strong electric fields accompany such plasma-enhanced chemical vapor deposition, which can enhance defect formation while complicating the analysis of growth processes. The dominant growth parameter under normal deposition conditions, however, is the substrate temperature, whose effect on the defect concentration is shown schematically in Fig. 9-2a. The minimum defect concentration near 3 ~ 1 0 cm-3 ' ~ is obtained for substrate temperatures between about 200 and 300 "C. Growth in this temperature range results in reasonably homogeneous, highly photoconductive, high resistivity a-Si :H films suitable for photovoltaic and photoelectronic applications. On either side of this optimal growth temperature range, the material properties degrade for reasons that depend on the details of the growth processes, which are not yet completely understood. In simple terms, the defect concentration increases as the growth temperature is reduced because strain-relieving chemical processes that depend on surface diffusion are kinetically limited. As the growth temperature

The electronic states specific to semiconductor surfaces and interfaces can affect device performance when the active layer are very thin. Although adsorption of water, oxygen, and other gases has been shown to lead to large changes in the measured conductivity of a-Si :H under certain conditions (Tanielian, 1982), the effects of surface or interface states on a-Si: H device performance are usually negligible. This is due, in part, to two factors. Firstly, there is little distinction between localized bulk and localized surface or interface states in terms of their distribution in the gap which might otherwise affect electronic transport and recombination processes. Secondly, the concentration of intrinsic surface states in a-Si :H is quite low: typically 5 x 10" cmW2 compared to 5 x 1014 cm-2 on c-Si. This is due to the effective passivation of the surface by hydrogen at growth termination which makes the a-Si :H surface relatively stable against oxidation in air compared to c-Si (Ley et al., 1981).

9.2.4.1 Surface States The concentration of intrinsic surface states has been determined by a variety of methods, usually by measuring the defect volume concentration as a function of film thickness and extrapolating to zero thickness (Jackson et al., 1983). The distribution of surface states has been measured by photoemission spectroscopy (Winer and Ley,

562


1987). Most methods find an intrinsic surface defect concentration between 1 and l o x 10" cm-' in optimally-grown intrinsic a-Si : H which increase upon n-type doping and initial oxidation. Though small by c-Si standards, this surface defect concentration is equivalent to quite a high volume defect concentration when the layer of a-Si:H is thin. This excess density of gap states at the surface relative to the bulk can be removed by slight boron doping. In fact, the lowest conductivity a-Si : H films are usually obtained by 10 vppm gas-phase boron doping which might be related to the removal of surface or interface states. The reduction in occupied surface gap states allows the valence band tail to be directly observed over a wide energy range by surface-sensitive photoemission spectroscopy. The exponential nature of the tail is maintained over several orders of magnitude in the density of states, and the slope of the tail agrees with values inferred from bulk transport and optical absorption measurements. In general, the distributions of localized surface and bulk states in the band gap are essentially the same. The surface properties of a-SiGe : H alloys are similar to those of a-Si : H (Aljishi et al., 1990), while little is known concerning the surface properties of a-Sic : H or other quaternary alloys systems. 9.2.4.2 Oxidation

In the initial stages of oxidation ( 0 1 monolayers), surface defect states are induced by activated oxygen adsorption which have been attributed to chemisorbed 3-fold coordinated surface oxygen atoms (Winer amd Ley, 1987). The distribution of these oxygen-induced states is similar to that of defects induced by n-type doping by overcoordinated phosphorus atoms incorporated from the plasma into the a-Si : H film

during growth. This suggests that the overcoordinated chemisorbed oxygen is likewise a surface donor, contrary to the normally strongly electronegative character of adsorbed oxygen. Upon further oxidation (for example, air exposure for long times) a true oxide layer forms with a change to slightly upward band bending and a correspondingly wider band gap than a-Si :H (Berner et al., 1987; Ley et al., 1981). In clean undoped a-Si : H, the free surface is under electron accumulation corresponding to an 0.5 eV downward band bending. Upon either n- or p-type doping, this downward band bending is removed. Activated oxygen adsorption pins the surface Fermi level just above midgap which results in an 0.25 eV downward band bending. On fully oxidized surfaces, however, the bands appear to be bent in the upward direction, as expected for the strongly electronegative surface oxide (Street et al., 1985b; Berner et al., 1987). 9.2.4.3 Interfaces

Interfaces are an integral part of all a-Si : H-based devices and their properties can greatly affect device performance. For example, the gate field that extends from the silicon nitride layer into the undoped a-Si : H active layer in an a-Si: H thin film transistor not only enhances the channel conductivity but can lead to defect creation near the a-SiN,: H/a-Si : H interface, which changes the transistor off voltage. Interface properties become even more important in multilayer structures. A multitude of phenomena initially observed in crystals have been observed in a-Si : H multilayer systems as well and generally result from the same physical origins. Examples are sub-band optical absorption (Hattori et al., 1988), resonant tunneling (Miyazaki et al., 1987), acoustic phonon


zone-folding (Santos et al., 1986), and persistent photoconductivity (Kakalios and Fritzsche, 1984). Multilayers of alternating n- and p-type a-Si:H layers (so-called “nipi” doping superlattices) and alternating alloy heterostructures (Le. a-Si :H/ a-Ge :H ...) have been grown and studied for many years. Such structures display a wide variety of phenomena similar to the crystalline analogs. After some initial uncertainties, there is now good evidence for carrier quantization in ultrathin a-Si :H heterostructures (Hattori et al., 1988; Miyazaki et al., 1987). Interface abruptness is a key problem for substantiating claims of quantized behavior in and exploitation of a-Si :H heterostructure multilayers. The low growth rates typical for a-Si: H (0.5-1 .O A/s, comparable to MBE growth rates) allow ultrathin multilayer growth with near atomic resolution (Yang et al., 1987). 9.2.5 Alloys

A major application of a-Si:H is for large-area solar cell arrays. In this application, the ability to lower the band gap of a-Si :H by alloying with Ge in order to more effectively match the solar spectrum has led to more efficient multijunction devices. On the other hand, alloying a-Si :H with carbon has led to efficient electroluminescent pin diodes with emission in the red, yellow, and green (Kruangarn et al., 1987). In addition, amorphous silicon nitrides (a-Si :N,: H) and oxides (a-Si : 0,: H) are used as gate dielectrics for TFTs, and for passivation and interlayer dielectrics in active matrix arrays. A-Si :H alloys are usually grown in the plasma-enhanced mode as pure a-Si :H but with the addition of methane, germane, ammonia, etc. as appropriate. The study of alloy properties has proceeded in parallel with that of a-Si: H. Our discussion is limited to weak alloys with the Group IV elements C

563

and Ge, since the properties of these are the most extensively studied. As might be expected these alloy properties deviate little from those of a-Si : Hat low alloy levels with a few interesting exceptions. Dilution of silane with methane for C incorporation generally slows the growth rate with a particularly steep drop occurring near 90% gas-phase methane mole fraction. This is due to the small incorporation probability of C for the PECVD growth mode at 250°C; the distribution coefficient for C under these conditions is typically less than 0.1. There is a large incorporation probability of Ge into a-SiGe:H films. However, growth of good quality a-SiGe :H is usually made at reduced growth rates, but this is due to the high hydrogen dilution which has been found to be necessary in order to reduce the otherwise large defect concentrations in undiluted a-SiGe :H films. Significant changes in the band gap of a-Si:H occur for C or Ge concentrations above about 10%. Infrared absorption spectra of such alloys show a Si-H stretching mode near 2100 cm-’ in addition to the stretching mode at 2000 cm-’ normally observed in pure, optimally-grown a-Si:H. The 2100 cm-’ mode increases in strength with increased C or Ge incorporation, similar to the increase observed with decreasing growth temperature in unalloyed a-Si:H. This behavior is believed to be due to increased hydrogen incorporation, perhaps in the form of clusters, which results from growth far from optimal conditions. The neutral defect (spin) concentration and Urbach edge (valence band tail) slope in undoped a-Si:H alloys also increase with increasing C or Ge incorporation. Post-deposition annealing near the deposition temperature can reduce the defect concentration by up to an order of magnitude with a concomitant decrease in the Urbach edge slope.

564


However, the infrared absorption spectra of a-SiGe: H and a-Sic : H alloys are unaffected by such thermal annealing. Alloy defect concentrations are always higher than that of unalloyed a-Si : H, usually by between 10 and 100 times. This is probably due to the inadequate knowledge of optimal growth conditions of a-Si : H alloys. Electron spin resonance data show that the dominant spinactive (g = 2.0055) defect in a-Si : H and its alloys are the same for moderate alloy compositions (Stutzmann et a]., 1989). The spin centers range to g = 2.0037 for C concentrations above about 40%, which probably is due to the emergence of C-C bonds as the dominant structural unit near this concentration. In a-Si:Ge:H alloys the Ge dangling bond dominates when the Ge concentration is above 20-30 at%.

9.3 Electronic Transport 9.3.1 Electrical Conductivity, Thermopower and Hall Effect The conductivity, a, is the product of the carrier density, n , and the carrier mobility, j&and the electron charge, e . (9- 10)

a=neb

The total conductivity is an integral over the density-of-states, N (E)

a = 5 N ( E ) e p (Elf@, T ) dE

(9-1 1 )

where f ( E , T ) is the Fermi function. The integral contains contributions from electron transport above the Fermi energy E, and hole transport below E,. When conductivity takes place far from E, by a single type of carrier. non-degenerate statistics can be applied and so (9-12) r

This equation is usually written as

-

-

o = - 1~ o ( E ) e x p l - ( E ~ ~ ) j d E (9-13) kT where o ( E ) = N (E) e p ( E ) kT

(9-14)

a(E) is the conductivity that would be observed when E, = E. The conductivity is determined by the density-of-states, the carrier mobility and the Boltzmann factor. When there is a sufficiently high defect density at the Fermi energy, as in unhydrogenated amorphous silicon, conduction takes place by variable range hopping, with a temperature dependence exp [-AIT”4] (Mott, 1968). The low defect density in a-Si : H prevents this mechanism from contributing significantly, and instead conduction takes place by electrons or holes at the band edges where both the density of states and the mobility increase rapidly with energy. For the particular case in which a(E) increases abruptly from zero at the mobility edge to a finite value ami,, energy, Ec, evaluation of Eq. (9-12) gives o ( T ) = ami,, exp [-(E,-E,)lkT]

(9-15)

omin is referred to as the minimum metallic conductivity and is given by a,,,=N(E,)ep,,kT

(9-16)

where is the free carrier mobility at E,. There is considerable doubt about the sharpness or even the existence of a mobility edge (Abrahams et al., 1979; Mott and Kaveh, 1985). Nevertheless, virtually all the conductivity experiments are analyzed in terms of Eq. (9- 15) which is a reasonable approximation even if p ( E )does not change abruptly, provided that it increases rapidly over a limited energy range. The conductivity of a-Si:H is usually thermally activated, at least over a limited

9.3 Electronic Transport

temperature range, and is described by,

o ( T )= ooexp [ - E , / k T ]

(9-17)

Comparison of Eqs. (9-15) and (9-17) suggests that a measurement of o ( T ) immediately gives the location of the mobility edge &-EF), and that the prefactor gives the conductivity at the mobility edge. In actual fact there is a huge variation in the values of oo which is shown by the results in Fig. 9- 15 for undoped and doped a-Si :H. The correlation between oo and the activation energy, E,, is referred to as the Meyer-Neldel rule (after its first observation in polycrystalline materials by Meyer and Neldel (1937)), and is described by

In oo= In ooo+ E , / k T ,

(9-18)

where ooois a constant with a value of about 0.1 S2-I cm-' and k T , is - 50 meV. A substantial part of the variation in the conductivity prefactor is due to the tempera-

71

I

I L *

A

6 7.7 -v

'C Y

D 0

0

.I5

-bo J 3

I

1

0

Q=*

I

I.o

0.5

Activation

Energy

E,, ( e V )

Figure 9-15. Measured values of the conductivity prefactor goversus the conductivity activation energy, showing the Meyer-Neldel rule (Tanielian, 1982).

565

ture dependence of the Fermi energy (Beyer and Overhof, 1984)

(E,

- EF) = (E, - E F ) O -

yT

(9-19)

from which it follows that

Various experiments show that oo is 100-200 52-' cm-', and that y can vary in the range ~ 5 x 1 eV/K, 0 ~ due to the shape of the density of states distribution and the temperature dependence of the band gap energy. The free mobility according to Eq. (9- 16) is &= oo/N(Ec) e k T = 10-15 cm2/Vs (9-21)

The thermopower measures the average energy of the transport with respect to the Fermi energy. The sign of the thermopower determines whether there is electron or hole transport, and confirms that n-type and p-type doping occurs in a-Si : H. This observation is important because the Hall effect, which is the more common measure of the doping type in crystalline semiconductors, has an anomalous sign in amorphous semiconductors as described shortly. The thermopower is expected to exhibit the same activation energy as the conductivity when conduction occurs above a well defined mobility edge (Cutler and Mott, 1969). Figure 9-16 shows examples of the temperature dependence of the thermopower in both ntype and p-type a-Si : H. The thermopower energy does decrease with doping, as expected from the shift of the Fermi energy by doping, but is always smaller than the conductivity energy by about 0.1 eV. This shows that the simple conduction model of an abrupt mobility edge is not exact, although the reason for the difference in energy of conductivity and thermopower is not resolved. Possible mechanisms include long range potential fluctuations of the band edges due to charged localized states, a

566

-

9 Material Properties of Hydrogenated Amorphous Silicon 1

I

l

1

Figure 9-16. Temperature dependence of the thermopower of n-type (left) and p-type (right) a-Si : H (see Fig. 10-9). The doping to levels range from 2 x lo-' (Beyer and Overhof, 1984).

I

\

-0.L

--0.8 ->

7 Y

E -1.2 ln

-1.6 L

L

0

1

L

-

u

d

J

2 3 - 4 103/T IK-'I

s

gradual rather than abrupt increase in the mobility near Ec, or polaron conduction. The charged defects and dopants in doped material make the potential fluctuations the most probable mechanism, but this model may not apply to undoped a-Si : H which has many fewer charged defects. A curious aspect of carrier transport in a-Si : H is the anomalous sign reversal in the Hall mobility coefficient. In doped a-Si : H the magnitude of the Hall mobility usually lies between 0.01 and 0.1 cm2Ns (LeComber et al., 1977) much lower than typical values (= 100-1000 cm*/Vs) observed in c-Si, and also lower than the drift mobility of a-Si:H. However, the sign of the Hall coefficient is opposite to the sign of the thermopower (i.e. positive in n-type and negative in p-type a-Si : H). Upon crystallization of a-Si:H, the Hall coefficients revert to their proper sign. The origin of this anomalous sign reversal is not completely understood, but is presumed to be related to the very short scattering lengths (Friedman, 1971: Emin, 1977).

lowed by excitation to the higher energy conducting states. The drift mobility, b,is the free carrier mobility reduced by the fraction of time that the carrier spends in the traps

9.3.2 The Drift Mobility

b = P D t

PQ = PO

(9-22)

When there is a single trapping level, with density N T , at energy E T below E,, b = iu, NcIIN, exp (ETlk T ) + N,] = = iu, ( N C I N T ) exp (-ETlk T )

(9-23)

where Nc is the effective conduction band density of states, and the approximate exh.The drift pression applies when b% mobility is thermally activated with the energy of the traps. A distribution of trap energies, N(E,), arising from the band tail of localized states, gives a drift mobility which reflects the average release time of the carriers. When the band tail distribution is sufficiently broad, the drift mobility becomes dispersive and is time dependent, following the relation (Scher and Montroll, 1975), 0

Conduction of electron and holes occurs by frequent trapping in the tail states fol-

tfree/ttrap

a-1

(9-24)

where a = TIT, ,

(9-25)

567

9.3 Electronic TransDort

is the dispersion parameter. The unusual time-dependence occurs because the probability that a carrier is trapped in a very deep trap increases with time. Of the many theoretical studies of dispersive transport by multiple trapping, the analysis of Tiedje and Rose (1980) and similarly Orenstein and Kastner (1981), is particularly instructive because the physical mechanism is easy to understand. The approach in this model is to consider an exponential band tail of traps with density proportional to exp (-ElkTo), where E is the energy from the mobility edge. A demarcation energy, E D , which varies with the time, t, after the start of the experiment is defined by ED

= k T In (mot)

1 VI

> \

E

u 10'

.-C x

= -.n

;10'

'c

6

10-

L

(9-26) 1

E D is the energy at which the average release time of the carrier from the trap is just equal to the timet. Provided the temperature is less than To, electrons in traps which are shallower than ED will have been excited to the mobility edge and trapped many times, but electrons in states deeper than E, have a very low probability of release within the time t. Thus the states deeper than ED are occupied in proportion to the density of states, but the states above E D have had time to equilibrate and follow a Boltzmann distribution for which the electron density decreases at smaller trap energies. The electron distribution therefore has a peak at E D , and from the definition of E,, this peak moves to larger trapping energies as time progresses. The approximation is made that all the electrons reside at ED, so that the problem can be treated as trapping at a single level, with the added property that the trap energy is time dependent. Substituting E, from Eq. (9-26) into Eq. (9-23), with the assumed exponential density of states gives ~(t>=~a(l-a)(mot)a-l

(9-27)

3

L

5

6

7

8

9

Inverse temperature 1 0 0 0 / T in K-'

1

7

lo-' m

> \

N

5

.-C

>r

c -.-n

10-2

E

c

c

&

10-~

2.2 2.6 3.0 3.4 inverse temperature 1 0 0 0 / T in

3.8

K-'

Figure 9-17. Temperature dependence of the (a) electron and (b) hole drift mobility at different applied fields ranging from 5 x 10' V/cm to 5 x 1O4 V/cm. The field dependence of pD is caused by the dispersion (Marshall et al., 1986; Nebel et al., 1989).

568


which has a power law time dependence in agreement with the measurements. In the time-of-flight experiment, the drift mobility is deduced from the time taken for a carrier to cross the sample. It is easily shown that the measured drift mobility is

1

pgp = O0 [ 2 p 0 ( l"- a >

lcli

(9-28)

where d is the sample thickness and F is the applied electric field. Thus the dispersive mobility is time, thickness and field dependent, and its magnitude is given in terms of the slope of the exponential band tail. Figure 9- 17 shows the field dependence of the electron and hole drift mobilities at different temperatures (Marshall et al., 1986, Nebel et al., 1989). The electron mobility is thermally activated, as expected for a trap limited process and at room temperature there is no field dependence and no dispersion because To S 300 K. However, in the low temperature dispersive regime, there is a large field dependence of and a time-dependent dispersion parameter a described by TIT,. The data for holes is qualitatively similar; the only difference is that the temperature scale is changed, so that the hole transport is dispersive up to 400500 K. The difference occurs because the valence band tail is wider than the conduction band tail. The mobility measurements find a dispersion parameter To=Tc of 250300 K for electrons and To=Tv of 400450 K for holes (Tiedje et a]., 1981).

9.4 Defect Equilibrium and Metastability Chemical bonding rearrangements are an important influence on the electronic properties of a-Si:H. Defect and dopant states

are created and annihilated either thermally or by external excitations such as illumination, leading to metastable structures. The thermal changes are described by thermodynamic equilibria with defect and dopant concentrations determined by minimization of the free energy. It is perhaps surprising to be able to apply equilibrium concepts to a-Si : H, because the amorphous phase of a solid is not the lowest free energy phase. However, subsets of network constituents may be in equilibrium with each other even if the network structure as a whole is not in its lowest energy state. The collective motion of many atoms is required to achieve long-range order, and there are strong topological constraints which usually prevent such ordering. However, chemical bonding transformations of defects of dopants require the cooperation of only a small number of atoms. Therefore, the small concentrations of defects or impurities in a-Si : H may be expected to participate in local thermodynamic equilibrium which takes place within the more or less rigid Si random network. Evidence that such equilibration is mediated by hydrogen motion is presented and discussed in Sec. 9.4.5.2.

9.4.1 The Hydrogen Glass Model The properties needed to describe the chemical bonding changes in a-Si :H are the equilibrium state and the kinetics of the process. Equilibrium is calculated from the formation energies of the various species, by minimizing the free energy, or equivalently, by applying the law of mass action to the chemical reactions describing the changes. For example, the reaction may correspond to a change in atomic coordination which causes the creation and annihilation of defect or dopant states Si,

* Si,

or

P,

* P,

(9-29)

Next Page

569

9.4 D e f e c t Equilibrium and M e t a s t a b i l i t y

The kinetics of the reaction are described by a relaxation time, t,, required for the structure to overcome the bonding constraints which inhibit the reaction. tR is associated with an energy barrier, E,, which arises from the bonding energies and is illustrated in Fig. 9-1 8 by a configurational coordinate diagram. The energy difference between the two potential minima is the defect formation energy, U,, and determines the equilibrium concentrations of the two species. The equilibration time is related to the barrier height by tR

=

exp (-E,lk T )

(9-30)

where w, is a rate prefactor of order 10' sec-' . A larger energy barrier obviously requires a higher temperature to achieve equilibrium in a fixed time. The formation energy U , and the barrier energy EB are often of very different magnitudes. There is a close similarity between the defect or dopant equilibration of a-Si : H and the behavior of glasses near the glass transition, which is useful to keep in mind in the analysis of the a-Si :H results. Configurations with an energy barrier of the type illustrated in Fig. 9-18 exhibit a high temperature equilibrium and a low temperature frozen state when the thermal energy is in-

sufficient to overcome the barrier. The temperature, TE, at which freezing occurs is calculated from Eq. (9-30) by equating the cooling rate, Rc, with dT/dt,,

TE In (w,k T&

EB) = E B / k

The approximate solution to Eq. (9-31) for a freezing temperature in the vicinity of 500 K is

k TE = EB/(30-1n R,)

(9-32)

from which it is readily found that an energy barrier of 1-1.5 eV is needed for TE= 500 K for a normal cooling rate of 10100 K/s. An order of magnitude increase in cooling rate raises the freezing temperature by about 40°C. Below TE, the equilibration time is observed as a slow relaxation of the structure towards the equilibrium state. Figure 9- 19 illustrates the properties of a normal glass by showing the temperature dependence of the volume, V,. There is a change of the slope of V o ( T )as the glass cools from the liquid state, which denotes the glass transition temperature TG. The

FROZEN IN STATE. (SLOWLY RELAXING)

,THERMAL EQUILIBRIUM

I TG

Barrier

(9-31)

W

/

91

/ >r

F C

w

,.' c-EQUILIBRIUM *.a*

z'

TEMPERATURE

Figure 9-18. Configurational coordinate diagram of the equilibration between two states separated by a potential energy barrier.

Figure 9-19. Illustration of the properties of a normal glass near the glass transition. The low temperature frozen state is kinetically determined and depends on the cooling rate.

Previous Page

570


glass is in a liquid-like equilibrium above TG, but the structural equilibration time increases rapidly as it is cooled. The glass transition occurs when the equilibration time becomes longer than the measurement time, so that the equilibrium can no longer be maintained so that the structure is frozen. The transition temperature is higher when the glass is cooled faster, and the properties of the frozen state depend on the thermal history. Slow structural relaxation is observed at temperatures just below TG. The glass-like characteristics are exhibited by the electronic properties of a-Si: H. However. a-Si:H is not a normal glass; it cannot be quenched from the melt. In a glass, all network constituents that contribute to the electronic structure participate in the structural equilibration. In a-Si : H, the disordered Si network is more or less rigid and the majority of Si atoms are fixed in a non-equilibrium configuration which persists up to the crystallization temperature (600°C). As is discussed in later sections, the kinetics of the equilibration of defects and dopants are governed by the motion of hydrogen, which mediates the coordination changes necessary in the approach to dopant or defect equilibrium. Virtually all hydrogen incorporated into the a-Si:H network participates in defect formation and dopant activation reactions, which are i n turn governed by a hydrogen chemical potential. It is the kinetics of hydrogen motion which determines the kinetics of defect formation and dopant activation reactions in a-Si : H. The analogy between dopant activation kinetics in a-Si : H and the kinetics of structural relaxation in glasses can be interpreted in terms of the glassy behavior of the hydrogen subnetwork in a-Si : H, which has been termed the hydrogen glass model (Street et al., 1987a).

9.4.2 Thermal Equilibration of Electronic States Figure 9-20 shows the temperature dependence of the d.c. conductivity of n-type a-Si : H for different thermal treatments, and the features of the data are obviously similar to those of glasses in Fig. 9-19. There is a change of slope at about TE= 130"C which distinguishes the high and low temperature regimes. Fast quenching from high temperature results in a higher conductivity than slow cooling, at a given temperature below 100°C. Above T,, the conductivity has a different activation energy and is independent of the thermal history. This is the equilibrium regime in which the defect and dopant densities are temperature dependent according to free energy minimization. The structure is frozen at lower temperature and has the metastable structure characteristic of a glass below the glass transition tempera-

1

2

I

3 4 INVERSE TEMPERATURE 1WOm IN K l

I

5

Figure 9-20. The temperature dependence of the d.c. conductivity of n-type a-Si : H, after annealing and cooling from different temperatures, and in a steady state equilibrium. The measurements are made during warming (Street et al., 1988b).

9.4 Defect Equilibrium and Metastability

ture. If the temperature is not too low, then there is slow relaxation of the structure which is described in Sec. 9.4.5. Metastable defects are also created thermally in undoped a-Si :H. Figure 9-21 shows the temperature dependence of the defect density between 200°C and 400 "Cfor material deposited under different plasma conditions. The defect density increases with temperature with an activation energy of about 0.2 eV. Although the defect density is reversible, a high metastable density is maintained by rapid quenching from the anneal temperature. Prolonged annealing at a lower temperature reduces the defect density back to its original value. Thus both doped and undoped a-Si:H have the glass-like property of a high tem-

700 650 600 I

1

TOW) 550 500 I

I

450

I

41

1

Y 3x10''

perature equilibrium and a low temperature frozen state. The equilibration temperature of undoped a-Si: H is higher than that of n-type material, indicating a slower relaxation process, arising from a higher barrier energy. The relaxation of p-type material is faster, yielding a lower equilibration temperature. Metastable defect creation is a related phenomenon. Here defects are created by an external stress such as illumination or bias. The defects are metastable provided that the temperature is well below the equilibration temperature, but are removed by annealing (see Sec. 9.4.6.).

9.4.3 The Defect Compensation Model of Doping The reversible changes in the conductivity of doped a-Si: H arise because the equilibration of defects and dopants alters the electrical conductivity. The charged defects act as compensating centers for the dopants, so that the density, n B T , of excess electrons occupying band edge states is lZBT = Ndon

-

571

- ND

(9-33)

where Ndonand ND are the dopant and defect concentrations. The conductivity is

*I

-gE-

10''

o = n B T e b

E

an

B

v)

\

U

::

0.18ev

3~10'~

1015

I

I

I

I

I

I

1.4 1.6 1.0 2.0 2.2 2.4 INVERSE QUENCH TEMPERATURE 1000/'T(r IN K-'

Figure 9-21. The temperature dependence of the equilibrium neutral defect density in undoped a-Si : H deposited with different deposition conditions (Street and Winer, 1989).

(9-34)

where ,q, is the effective drift mobility. Thus, changes in the density of donor and defect states are reflected in the conductivity. Equilibration occurs between the different bonding states of the silicon and dopant atoms, which can both have atomic coordination 3 or 4.The lowest energy states are Si! and Pi, as indicated by the 8-N rule (see Sec. 9.1.3). The formation energies of P, and Si, are large enough that neither would normally be expected to have a large concentration. However, when both states are formed, the electron liberated from the donor is trapped by the dangling bond, lib-

572


erating a substantial energy and promoting their formation. The compensation of phosphorus donors by defects is described by the chemical reaction

P!

+ Si," * P: + Si,

(9-35)

The equilibrium state of doped a-Si:H is calculated by applying the law of mass action Np, = NS13= K NoNpj ; K = exp [ - ( u p + U D ) / k TI

(9-36)

where the different N's denote the concentrations of the different species, No is the concentration of 4-fold silicon sites, and K is the reaction constant. Up and U D are the formation energies of donor and defect. Single values of the formation energies are assumed for simplicity: the next section includes a distribution of formation energies. which is more appropriate for a disordered material. When the doping efficiency is sufficiently low that N p 44 N P 3 ,and N p , is equated to Ns,, as required by charge neutrality, then Eq. (9-36) becomes NP, = NSI7 = = (NoNp)"' eXp

[-(up+

(9-37) l / ~ ) / k2 TI

This equation predicts the square root law for defect creation which is observed in the data of Fig. 9- 12. The thermodynamics also predicts that the doping efficiency is temperature dependent and explains the high metastable d.c. conductivity which is frozen in by quenching (Street et al., 1988a). Defect equilibration also occurs in undoped a-Si : H. In the absence of dopants, the model predicts that the temperature dependence of the defect density is NDO = No exp (-l/Ddk T )

(9-38)

The defect density does increase with temperature as is seen in the data of Fig. 9-21.

However, the temperature dependence has an activation energy of only about 0.2 eV and a small No, whereas the model for doping just described indicates a considerably larger value of the formation energy (Street et al., 1988a). The difference originates from the distribution of formation energies, which must be included to get the correct defect density. This is discussed in the next section. This type of defect reaction provides a general explanation of all the other metastable phenomena described in Sec. 9.4.6. The formation energies of charged defects and dopants depend on the position of the Fermi energy, E, defect:

UD = U D o - (EF-

ED)

(9-39)

dopant: Up = Upo - (Ep- EF)

(9-40)

UDoand Upo are the formation energies of the neutral states and ED and EPare the associated gap state energies. The second terms in Eqs. (9-39) and (9-40) are the contributions to the formation energy from the transfer of an electron from the Fermi energy to the defect or from the donor to E,. The negatively charged defect density is given by a Boltzmann expression (Shockley and Moll, 1960)

and there are similar expressions for positive defects and donors. Equation (9-41) assumes that the defects have the same formation energy and gap state levels in doped and undoped a-Si : H. The defect density is, therefore, a function of the position of the Fermi energy, and Eq. (9-41) expresses the interaction between the electronic properties and the bonding structure. The equilibrium defect density in Eq. (9-41) increases


exponentially as E, moves from the dangling bond gap state energy. Thus, doping increases the defect density, as does any other process which moves the Fermi energy from mid-gap, such as illumination or voltage bias. The doping efficiency is suppressed by doping, but enhanced by compensation. All of these effects are observed in a-Si :H.

(a) Conduction band

,-*u

r

'*, bond /

8 0

4

ED

/

I

0

Valence band

(b)

where No (UD) is the distribution function of the formation energy. Calculations of the distribution of formation energies have been addressed by the weak bond model (Stutzmann, 1987; Smith and Wagner, 1987). Figure 9-22a shows a schematic model of a weak Si-Si bond, and a pair of dangling bonds. When the weak bond is converted into a neutral dangling bond, the electron energy increases from that of a bonding state in the valence band to that of a non-bonding state in the gap. The formation energy of a neutral defect is

-E,,

Dangling

'\

Weak bond

The random network of an amorphous material such as a-Si : H implies that the formation energy varies from site to site. A full evaluation of the equilibrium must include this distribution and also the disorder broadening of the defect energy levels. The generalized form of Eq. (9-38) is

U D O = ED

.

\

-V

9.4.4 The Weak Bond Model

573

+

Weak bond

Figure 9-22. (a) Energy level diagram showing the conversion of a weak Si-Si bond into a dangling bond: (b) Illustration of the hydrogen-mediated weak bond model in which a hydrogen atom moves from a Si-H bond and breaks a weak bond, leaving two defects (DHand D,) (Street and Winer, 1989).

trons, before and after the bond is broken and the sum represents the change in energy of all the valence band states other than is the from the broken weak bond. Mion change in energy of the ion core interaction for the structure with and without the defect. The weak bond model assumes that the terms in the square bracket in Eq. (9-43) are small, so that uDO

1

E , and EWBare the gap energies of the defect electron and of the valence band tail state associated with the weak bond. E, and E; are energies of the valence band elec-

E

ED-

(9-44)

The distribution of defect formation energies is therefore described by the density of valence band tail states. Most states have a high formation energy, but an exponentially decreasing number have lower formation energies. In equilibrium, virtually all the band tail states which are further from the

574


valence band than ED convert into defects, while only a small temperature dependent fraction of the states between E , and the mobility edge convert. The weak bond model is useful because the distribution of formation energies can be evaluated from the known valence band and defect density of states distributions. A calculation of the defect density requires a specific physical model for defect creation. Dangling bond defects form by the breaking of silicon bonds, and several specific models have been proposed (Smith and Wagner, 1987; Street and Winer, 1989; Zafar and Schiff, 1989). We analyze a model in which the bonds are broken by the motion of hydrogen. Figure 9-22 b shows hydrogen released from an Si-H bond, breaking a Si-Si bond to give two separate defects. Experimental evidence for the involvement of hydrogen in the equilibration is described in a later section. The hydrogen-mediated weak bond model of Fig. 9-22 b is described by the defect reaction Si-H

+ (weak bond) * DH + Dw

(9-45)

The two defects may be electrically identical but make different contributions to the entropy. The law of mass action solution for the defect density, including the distribution of formation energies, is (Street and Winer, 1989) ND = 2 N HNvo.exp(-ED/kTv). lk . I NHexp+ ND exp(2UDo/kT) TV) d u D O

(9-46)

where T, is the slope of the exponential valence band tail. NVois the band tail density of states at the assumed zero of the energy scale for E,. Numerical integration of Eq. (9-46) gives an excellent fit to the data of Fig. 9-2 1, for band tail and defect parameters which are consistent with the known electrical properties. The weak temperature

dependence of N , follows directly from the distribution of formation energies (Street and Winer, 1989). The weak bond model also explains the variation of the defect density with the growth conditions in the plasma reactor. In material with more disorder, the valence band tail is broader (i.e. larger Tv) and ND increases according to Eq. (9-46). The defect density is conveniently expressed as ND = E (T, Tv, NH) Nv, k Tv * exp [-E,/k Tv]

*

(9-47)

where 6 (T, Tv, NH)represents the entropy factor which differs for each specific defect creation model, but which is a slowly varying function. The equilibrium defect density is primarily sensitive to Tv and E,, through the exponential factor. For example, raising T, from 500 K to 1000 K increases the defect density by a factor of about 100. The sensitivity of the defect density to the band tail slope accounts for the large change in defect density with deposition conditions and annealing. Figure 9-23 shows the correlation between the valence band tail slope and the defect density for undoped a-Si : H deposited by different methods and under different deposition conditions. The data show that a high defect density is correlated with a wide band tail slope, and is explained by the equilibrium model. The band tails are much broader at low deposition temperatures, so that Eq. (9-47) predicts the higher defect density which is observed. The defect density is reduced when the low deposition temperature material is annealed, and the band tail slope is correspondingly reduced. Similarly, the slope of the Urbach tail and the defect density both increase at deposition temperatures well above 300°C, and are associated with a lower hydrogen concentration in the film. Both E, and T, may change in alloys of a-Si : H and this is perhaps the origin of the differ-

9.4 Defect Equilibrium and Metastability I

0

I

50 100 Band tail slope E,, in meV

150

Figure 9-23. Dependence of the defect density on the slope of the Urbach absorption edge for undoped a-Si :H deposited under a variety of conditions (Stutzmann, 1989).

ent defect densities in these materials. There is a larger defect density in low band gap a-SiGe : H alloys, which is predicted from the reduced value of ED accompanying the shrinking of the band gap. In the larger gap alloys, such as a-Sic :H, the predicted reduction in N D due to the larger ED seems to be more than offset by a larger Tv, so that the defect density is again greater than in a-Si:H. There is, however, no complete explanation of why a-Si: H has the lowest defect density of all the alloys which have been studied.

9.4.4.1 The Distribution of Gap States The theory described above only considers defects with a single energy in the gap. Neutral defects in undoped a-Si:H are known to be distributed in an approximate-

575

ly Gaussian band -0.1 eV wide. Bar-Yam and Joannopoulos et al. (1986) first pointed out that the minimization of the free energy of the broadened defect band causes a shift of the defect gap state energy level. The reason is that the gap state energy is contained within Eq. (9-44) for the defect formation energy. Thus, states that are at a lower energy in the band gap will have a lower formation energy and, therefore, a higher equilibrium density. This is the basis of the defect pool concept, in which there is a distribution of available states where defects can be formed, which are selected on the basis of energy minimization. The interesting feature of this dependence on gap state energy is that it leads to different defect state distributions depending on the charge state of the defect. The formation energy of positively charged defects is not influenced by the energy of the gap state, because the defect is unoccupied. On the other hand, negative defects contain two electrons and the gap state energy enters twice. For a Gaussian distribution of possible defect state energies, there is a shift of the defect band to low energy by (Street and Winer, 1989) B2

AE=k TV

(9-48)

for each electron in the defect, where B is the width of the Gaussian distribution. Measured values of the defect band width are imprecise but lie in the range 0.20.3 eV, corresponding to B = 0.1 eV. The predicted shift of the peak is therefore about 0.2 eV, when k T, = 45 meV. This shift of the defect energy with doping explains some of the differences in the measured defect energies. The equilibration process has the effect of removing gap states from near the Fermi energy. Either unoccupied states above EF, or occupied states below E, are

576


energetically preferred to partially occupied states at EF. The shift of the defect levels with doping is illustrated in Fig. 9-14.

9.4.5 Defect Reaction Kinetics

I

The relaxation kinetics follow a stretched exponential relation AnBT

There is a temperature dependent equilibration time associated with the chemical bonding changes. The very long time constant at room temperature is responsible for the metastability phenomena, because the structure is frozen. An example of the slow relaxation towards equilibrium in n-type a-Si:H is shown in Fig. 9-24 by the time dependence of the electrons occupying shallow states, n B T , following a rapid quench n B T decays slowly to a steady from 21O0CC. state equilibrium, with the decay taking more than a year at room temperature, but only a few minutes at 125 ' C . The temperature dependence of the relaxation time, 7, is plotted in Fig. 9-25, and has an activation energy of about 1 eV, which measures the energy barrier for bonding rearrangement. The relaxation is faster in p-type than in n-type a-Si : H and has a slightly lower activation energy. A similar relaxation occurs in undoped a-Si : H, with a larger activation energy and a longer relaxation time (Street and Winer, 1989). Equations (9-30) and (9-3 1) relate the relaxation time to the equilibration temperature.

I

9.4.5.1 Stretched Exponential Decay

I

I

= no eXp [ - ( f / r ) P ]

(9-49)

with O < p < 1 (Kakalios et al., 1987). This type of decay also describes the equilibration of a wide class of disordered materials, particularly the structural relaxation of a normal glass. The origin of the stretched exponential is the disorder which leads to a distribution of activation barriers for structural change. However, the mathematical description of the relaxation is complicated and is not completely resolved. The relaxation may be understood from a simple rate equation with a time dependent rate constant, k (r). dddt = - k ( t ) TI

(9-50)

The time dependence of k ( t )reflects a structural relaxation occurring within a system which is itself time dependent. If k has a time dependence of the form k ( t ) = ko

(9-51)

tP-'

then the integration of Eq. (9-50) gives the stretched exponential n ( i ) = no exp [ - ( r / t > P ]

I

I

n-type

(9-52)

I

a-Si H

-

-

02-

1

10

io2

lo3

10'

Time in s

lo5

io6

10'

io8

Figure 9-24. The timedependent relaxation of the band tail carrier density plotted in normalized form. The solid lines are fits to the stretched exponential with parameter p as indicated (Kakalios et al., 1987).


577

The next section shows that the diffusion properties of hydrogen within a-Si :H quantitatively account for the relaxation rates and the stretched exponential decay. io5

9.4.5.2 Hydrogen Diffusion

The hydrogen diffusion coefficients shown in Fig. 9-26 are thermally activated

D, = Do exp ( - E H D / k T )

Inverse temperature 1000/ T in K-'

Figure 9-25. Temperature dependence of the relaxation time in n-type and p-type a-Si :H (Street et al., 1988a).

The temperature dependence of the parameter p, measured from the relaxation data of Fig. 9-24, follows the relation

p = T/TE

T, = 600 K

(9-53)

The equilibration rate is determined by the motion of either silicon, hydrogen or the dopants, which cause the structural relaxation. Equation (9-5 1) implies that the atomic motion has a power law time dependence. Hydrogen is known to diffuse fairly easily in a-Si : H, as is obvious from the fact that it is evolved from a-Si :H at a fairly low temperature (400-500°C). On the other hand, the silicon network is rigid and stable, and both silicon and the dopants have a much lower diffusion rate. Hydrogen motion is therefore an obvious candidate for the mechanism by which the thermal equilibration takes place.

(9-54)

where the energy E,, is about 1.5 eV, and cm2 s-'. The the prefactor Dois about diffusion is also quite strongly doping dependent, as seen in the data for p-type, ntype and compensated material. The greatest change is with the p-type material for which D, is larger by a factor lo3 at 250°C. The increase is less in n-type material, while the diffusion rates in compensated and undoped a-Si :H are similar. The low diffusion in compensated material suggests that the doping effect is electronic rather than structural in origin. The diffusion activation energy is similar to the energy for structural relaxation, and the trends with doping mirror the doping dependence of the equilibration rates (compare Figs. 9-25 and 9-26). Thermally-activated diffusion is explained by a trapping mechanism. There is no significant diffusion of silicon at these temperatures, so that hydrogen diffusion occurs by breaking a Si-H bond and reforming the bond at a new site. Hydrogen is trapped by bonding to the silicon dangling bonds and the mobile hydrogen moves between higher energy interstitial sites. Highly distorted Si-Si bonds also act as traps for hydrogen which can break the weak bond by the reaction shown in Fig. 9-22. The interstitial is the Si-Si bond center site in crystalline silicon and is most probably the same in a-Si: H (Johnson et al., 1986). For this model the diffusion is given by

D, = ak wo exp ( - E , d k T )

(9-55)

578


0

5 -

2

4 -

t

-5 I-

DEUTERIUM DIFFUSION 10-2 BzHs/SIH4 200'C

3 -

Y

"E lo-"u C ._

0

2 -

Y

$ z

0'

0

.t

E

g1x101sVI 0.8 -

r

k

._ 10-6-

.-V

0

0.6-

0 C

;. 10-'63 r ._ TI r

undoped ICorlson and Magee)

C

0

e 10'"-

P x

I

10-'1

, 1.6

I

1.8

\10-3~,10-3~

inverse temperature IOOO/

1

2.2

2.0

r in K-'

Figure 9-26. The temperature dependence of the hydrogen diffusion coefficient at different doping levels is indicated (Street et ai., I987 b), including data from Carlson and Magee ( 1978). Lower lines: undoped, o compensated.

where w, s-' is the attempt-to-escape frequency for the excitation out of the trapping site, U , is the distance between sites, and E,, is the energy to release hydrogen from a Si-H bond. A diffusion prefactor of lo-* cm2 s-' results from a hopping distance of about 3 A. However, the prefactor is difficult to interpret because there are other hydrogen traps in addition to the dangling bonds and the measurement is influenced by the time-dependence of the diffusion. The hydrogen diffusion coefficient is not constant, but decreases with time (Street et al., 1987b). The data in Fig. 9-27 show a power law decrease in p-type a-Si : H of the form t B - ' , with /3 = 0.8 at the measurement temperature of 200°C. The time dependence is associated with a distribution of traps originating from the disorder. The ef-

fect is similar to the dispersive transport of electrons and holes in an exponential band tail. The time dependence of the hydrogen diffusion is of the form of Eq. (9-51), and accounts for the stretched exponential decay, since the temperature dependence of the diffusion dispersion parameter agrees with the value from the relaxation measurements. These results indicate that the motion of hydrogen is the rate limiting process in the bonding equilibration. Hydrogen can change the coordination of a silicon or impurity atom by breaking and forming bonds, causing the creation and annihilation of defects and dopants.

9.4.6 Metastability A serious limitation to some applications of a-Si : H is the phenomenon of reversible metastable changes which are observed in the electronic properties when stress is applied. Such changes are induced by illumination, voltage bias or an electric current, and are reversed by annealing to 150200°C. The most widely studied metastability is the creation of defects by prolonged illumination, known as the Staebler-Wronski effect (Staebler and Wronski, 1977). In

579


addition, space-charge-limited current flow in p-i-p structures induces defects (Kruhler et al., 1984), and studies of the current-voltage characteristics of thin film transistors find a threshold shift which is due to defects in the a-Si :H layer, induced by the electron accumulation at the interface (Jackson and Kakalios, 1989). In common with all the metastable changes, the annealing process is thermally activated with an energy about 1 eV, so that annealing of undoped a-Si:H takes a few minutes at 200"C, several hours at 150"C, and an indefinitely long time at room temperature. The distinctive metastable and equilibrium structural changes in a-Si :H arise from interactions between the electronic configuration and the atomic structure. Further evidence that hydrogen plays a key role in this structural change, as proposed by the hydrogen glass model, is the observation that hydrogen diffusion is enhanced by illumination. This effect was reported by Santos et al. ( 1 991) for undoped a-Si :H at temperatures near 200 "C. Figure 9-28 shows SIMS profiles of deuterium diffusion, comparing the results with and without white light illumination. The diffusion coefficient is enhanced by a factor of 2-5, and the enhancement is approximately linear with the light intensity. The explanation of this enhancement is that recombination of excess electron-hole pairs excites hydrogen to a mobile state. Along with the increase in diffusion coefficient, the mobile hydrogen also creates metastable defects, as described by the specific models discussed in the next section. The external excitation may change both the reaction rate and the equilibrium state of a defect reaction. Figure 9-29 illustrates the expected temperature dependence of the reaction rate in different situations. The activation energy of a purely thermal defect creation process is E,, which is about 1 eV

a) W0C,17 h

I

I

0 200 Depth Mo rSiH (nm)

400

Figure 9-28. Deuterium concentration profiles for diffusion in the dark (thin lines) and under an illumination intensity of 17 W/cm2 (thick lines) at different temperatures and times as indicated. Curve (d) is for diffusion from a deuterium plasma (Santos et al., 1991).

V.9

Ev

I

Inverse temperature in K-'

Figure 9-29. The expected temperature dependence of the reaction rate for a recombination-enhanced defect creation process (Kimerling, 1978).

580


in a-Si:H and associated with hydrogen diffusion. An external excitation can provide a non-thermal source of energy to overcome the potential barrier - the non-radiative recombination of excess carriers is an example. The recombination-enhanced reaction rate has a lower activation energy of EB - E x ,where Ex is the energy provided by recombination (Kimerling, 1978).The reaction rate is athermal when E, is equal to or larger than E,, but is thermally activated when E , is smaller. Defects or dopants are created because the external excitation drives the defect reaction away from the initial equilibrium. The formation energy of charged defects, and, therefore, their equilibrium density, depends on the position of the Fermi energy according to Eq. (9-39). Thus, the equilibrium defect density is N,,

= N D B o exp (AE,lk T )

(9-56)

where AEF is the shift of the Fermi energy, and NDBo is the neutral defect density corresponding to an unshifted Fermi energy. Any external stress which causes the Fermi energy to move disturbs the equilibrium of the states and tends to change the defect density. The examples of metastability discussed below cover most of the possible ways in which the Fermi energy (or quasiFermi energy) can be moved by an external influence. Others include bulk doping (Sec. 9.4.3) and oxidation (Sec. 9.2.4.2).

9.4.6.1 Defect Creation by Illumination The most widely studied example of metastability in a-Si : H is the creation of defects by prolonged exposure to light. The defects are dangling silicon bonds and are detected by ESR and other defect spectroscopy techniques. Figure 9-30 shows the time dependent increase in the defect density, which follows a r1'3 law (Stutzmann et al., 1985).

ILLUMINATION TIME (min)

Figure 9-30. The time dependent increase of the defect density by illumination of undoped a-Si : H for different illumination intensities. The lines show the fit to a (time)"3 dependence. The intensity dependence at constant illumination time is shown in the insert (Stutzmann et al., 1985).

The defect density increases from 10l6 cm-3 to about 10'' after an hour of sunlight illumination. Although Fig. 9-30 shows no sign of saturation after 2-3 hours of strong illumination at room temperature, a steady state is obtained at higher light intensities or longer illumination times (Park et al., 1990). The defect creation results from the non-radiative recombination of electrons and holes, which releases enough energy to cause bonding rearrangement. The sub-linear creation kinetics occurs because the additional defects act as recombination centres and suppress the density of excited carriers. The excess defects are metastable and are removed by annealing above about 150°C, so that the defect creation process is reversible. Defects are created by the recombination of photoexcited carriers, rather than by the optical absorption. The experimental evidence is that defect creation also results from charge injection without illumination, and that defect creation by illumination is suppressed by a reverse bias across the sam-


ple which removes the excess carriers (Swartz, 1984). The defect creation rate is almost independent of temperature, indicating that all the energy needed to overcome the barrier is provided by the recombination. The kinetics of defect creation are explained by a recombination model which assumes that the defect creation is initiated by the non-radiative band-to-band recombination of an electron and hole. The recombination releases about 1.5 eV of energy which breaks a weak bond and generates a defect. In terms of the configurational coordinate model of Fig. 9- 18, the recombination energy completely overcomes the barrier E,. Stutzmann et al. (1985) propose the following model to explain the defect creation. The creation rate is proportional to the band-toband recombination rate dND/dt = Cd n p

(9-57)

where cd is a constant describing the creation probability, and n andp are the electron and hole concentrations. The defect creation process represents only a small fraction of the recombination, most of which is by trapping at the dangling bond defects. The recombination is quite complicated, but provided the illumination intensity G is high enough, the carrier densities n and p at the conduction and valence band edges are given to a fair approximation by

n = p = G/A N,

(9-58)

where A is an average recombination constant. Photoconductivity experiments, which measure the carrier concentration n, confirm the predicted dependence on G and ND, although at low defect density there are deviations from a monomolecular decay. The time dependence of the defect density is obtained by combining Eqs. (9-57) and (9-58) and integrating, to give Nh ( t ) - Nh (0) = 3 Cd G 2 t/A2

(9-59)

581

where ND(0) is the initial equilibrium defect density. At sufficiently long illumination times such that N,(t) >2ND(0), Eq. (9-59) approximates to (9-60) which agrees well with the intensity and time dependence of the experimental results. Recently, Branz (1998) proposed an alternative model for defect creation which has essentially the same kinetics. Instead of the electron-hole pair recombination being the rate-limiting step in the generation of metastable defects, Branz proposes that the pairing of excited mobile hydrogen atoms determines the kinetics. Rather than a single interstitial hydrogen atom breaking a weak bond, as illustrated in Fig. 9-22, it is postulated that two hydrogen atoms must come together at a weak bond to stabilize the configuration, otherwise the single hydrogen will diffuse away. Since the bimolecular binding of hydrogen essentially yields the same defect creation kinetics as for the Stutzmann model, other measurements are needed to decide between them. The annealing kinetics of the light induced defects are shown in Fig. 9-3 1. Several hours at 130 "C are needed to anneal the defects completely, but only a few minutes at 200°C. The relaxation in non-exponential, and in the initial measurements of the decay the results were analysed in terms of a distribution of time constants (Stutzmann et al., 1986), which is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential. The parameters of the decay the dispersion, p, and the temperature dependence of the decay time r, - are similar to those found for the thermal relaxation data, and so are consistent with the same mechanism of hydrogen diffusion mediating the chemical reactions. The annealing is

582


I

I

-

-

Figure 9-31. The decay of the normalized light induced defect density at different anneal temperatures, showing the stretched exponential behavior (Jackson and Kakalios, 1989).

- O O

1

10 Time in min

therefore the process of relaxation to the equilibrium state with a low defect density.

9.4.6.2 Defect Creation by Bias and Current Similar defect creation occurs when excess space charge is introduced in undoped a-Si:H. This is achieved by biasing into electron accumulation the interface of a-Si : H with a dielectric. Figure 9-32 shows the time dependent change in the threshold voltage of the junction capacitance, which is proportional to the induced defect density. In this case, as many as lo'* cm-3 defects are created over an extended time. The defect creation rate is thermally activated, and the rate increases by a factor 100 between 360 K and 420 K. The defect creation kinetics follow a stretched exponential time dependence with parameters which are similar to those found for the equilibration kinetics discussed in Sec. 9.4.5.1. The defect creation rate is low at room temperature but is thermally activated with an energy of about 1 eV, and has a time constant of a few minutes at 150°C. A space charge limited current flow in a pc-p-pc structure also results in a metastable increase in the defect density (Kruhleret al., 1984). The presence of extra defects is

100

1000

inferred from the reduction in the current seen in the current-voltage characteristics. As with the other metastable changes, the effect is reversible by annealing to about 150°C. Both of these experiments show that electron-hole recombination is not necessary to create defects, since there are virtually no minority carriers in either case. The energy needed to form the defects must instead come from one carrier.

100

10'

10'

106

Time in s

Figure 9-32. Time dependence of the defect density created near a dielectric interface by voltage bias. The measurement is of the threshold shift, AV, which is proportional to the defect density (Jackson and Kakalios, 1989).

9.5 Devices and Applications

9.5 Devices and Applications The various applications of amorphous silicon take advantage of its ability to be deposited on large area substrates. During the 1970s and 1980s, the main application was for solar power generation (Carlson and Wronski, 1976). The single p-i-n diode structure was developed first, and great care was taken to optimize the cell efficiency by using all the knowledge gained about electronic transport and defects in a-Si:H and by choosing the thickness of the Iayers carefully. Multi-junction cells followed because of the higher theoretical efficiency, and resulted in cells with two or three stacked p-i-n diodes, using different alloy materials to capture the maximum amount of the solar spectrum (Guha, 1996). The invention of the a-Si:H thin film transistor (TFT) (Le Comber et al., 1979) opened up possibilities for large area electronic devices based on matrix addressing. Linear scanners for FAX machines were made first, but the most significant product has been the active matrix liquid crystal display (AMLCD) (Snell et a]., 1983; Tsukada, 1999). This is now a $ 20 billion industry for lap-top computers, and is expected to become even more widespread for desk-top computer monitors and flat panel TVs. Active matrix image sensors have recent emerged as an important technology for Xray medical imaging because of their high sensitivity and compactness (Street and Antonuk, 1993; Street, 1999). Along with photoreceptors and xerography (Pai, 1988), position sensors (Martins and Fortunato, 1999), integrated color sensors (Palma, 1999), light-emitting diodes (Krunagam et al., 1987), and sensors for CMOS (complementary metal oxide semiconductor) video cameras (Bohm, 1998), the set of a-Si :H applications is expanding rapidly.

583

The key a-Si: H devices which makes these applications possible are the thin film transistor (TFT) and the p-i-n photodiode. TFTs provide switching for active matrix addressing, and the diodes provide sensing and photovoltaic energy conversion. These two devices are described first, before a more detailed discussion of the active matrix arrays.

9.5.1 Thin Film Transistors The usual a-Si :H TFT structure, as illustrated in Fig. 9-33, is a bottom gate device with a silicon nitride gate dielectric, an undoped a-Si : H channel, n-type a-Si : source and drain contacts, and silicon nitride passivation. This structure takes advantage of the available materials with a common deposition technique, and applies the knowledge that has been gained about doping and alloying of a-Si : H, as discussed in Sec. 9.2. The bottom gate structure develops less charge trapping than a top gate design, partly because a low defect density silicon nitride alloy can be deposited at about 350 "C without affecting the a-Si :H layer which is usually deposited at a lower temperature. The TFT operates in the electron accumulation mode, again a result of the particular material properties of the a-Si : H described earlier. Electrons have much higher drift

length L

,,#,T

e---------->

L

L

gate insulator Igatecontaci

I

Figure 9-33. Bottom gate TFT structure with source and drain contacts overlapping the channel.

584


mobility than holes, and the high defect density of doped a-Si : H precludes the use of a doped channel operating in depletion mode. All of the thin films that comprise the TFT can be deposited by the same PECVD deposition technique in reactors that are relatively easy to scale to large area. In the most common process used for the device of Fig. 9-33, the source-drain contacts are deposited over the protecting nitride passivation layer which is self-aligned to the gate. The overlap of the doped source and drain contacts over the TFT channel is necessary to ensure low resistance contacts, but provides some parasitic coupling capacitance to the gate which can have a significant detrimental effect on the speed and performance of matrix-addressed arrays. Fully self-aligned TFTs are being developed to minimize the capacitance (Powell et al., 1998). The T I T characteristics approximate well to the usual MOS transistor relations for linear and saturation behavior, and an example of TFT transfer characteristics is shown in Fig. 9-34. In the linear regime, the on-resistance, R O N ,is given by the ratio of the source-drain voltage and current,

where V, is the gate voltage, V, the threshold voltage, W the TFT width, and L the length. The typical nitride thickness of 3000 A gives a gate capacitance, C,, of 5 x 1 O-' F cm-*, and the field effect mobility, pFEis 0.5-0.8 cm2 V-' s-'. These values yield TFTs with an on-resistance of a few megaohms for a small width-to-length ratio. WIL 1 - 5 and VG- V,- 10 V, and they are well suited to the requirements of active matrix arrays. The minimum leakage current shown in Fig. 9-34 corresponds to an extremely large off-resistance (- 10ls which also turns out to be important for active matrix arrays.

-

a),

3 1E-08

8

e

$ 5w

1E-10 1E-12

0

1E14 1E16 -10

0

10

20

sourcedrain voltage M Figure 9-34. The typical TFT transfer characteristics showing low leakage and a high on/off ratio.

The leakage current is probably a bulk thermal generation current in the channel, similar to the mechanism of photodiode leakage described below. Also contributing to the low current are the barrier formed by the n-type source and drain contacts, the low density of states at the a-Si : Hhitride interface, and the low carrier mobility of a-Si : H. The leakage current is considerably smaller than in polysilicon devices, so that the a-Si : H device is preferred for many matrixaddressed arrays, even though the polysilicon device has higher mobility. In the subthreshold region with V, below V,, the current increases exponentially due to the presence of the band tail localized states.

9.5.2 P-i-n Photodiodes and Solar Cells The typical p-i-n photodiode design is illustrated in Fig. 9-35 and consists of thin p-doped and n-doped a-Si : H layers separated by a thicker undoped (i) layer. The doped layers provide rectifying contacts but contribute little to the light sensitivity, and so are made as thin as possible. Section 9.2.3.2 describes how doping causes a high density of charged dangling bond defects in a-Si : H: positive in n-type material and neg-


Figure 9-35. Illustration of the p-i-n photodiode structure showing the thick (- 1 pm) undoped layer and very thin doped contact layers.

ative in p-type material. The minority carrier lifetime in doped a-Si : H is therefore so small that most holes generated in the nlayer and electrons in the p-layer recombine before they can cross the reverse bias junction. On the other hand, the high defect density provides a very small depletion width, so that a doped layer thickness of 100 A or less is sufficient to form the junction.

9.5.2.1 The Photodiode Electrical Characteristics The standard analysis of semiconductor diodes gives a forward current of (9-62) where n is the ideality factor. Figure 9-36 shows that a-Si : H p-i-n diodes exhibit the predicted exponential forward current over several orders of magnitude, with an ideality factor of about 1.5. According to the theory of crystalline semiconductors, the transport mechanism of thermionic emission over the barrier yields n = 1, while bulk generation-recombination gives n = 2. As with many other aspects of the transport in a-Si : H, a full understanding of the diode

585

data requires numerical modeling. Approximate analytical expressions are usually not accurate enough, because of the continuous distribution of localized states in the gap. Figure 9-36 compares the measured and calculated forward bias current, and shows the effects of increasing defect density (Street and Hack, 1991). Both the ideality factor and the changes with high defect density are predicted correctly by the model, which assumes that transport is dominated by bulk recombination currents, with insignificant contributions from thermionic emission. This contrasts with the behavior of metal Schottky contacts on a-Si : H (e.g., platinum, palladium, etc.), for which the ideality factor is close to unity and the mechanism is thermionic emission over the barrier (Thompson et al., 1981). In a-Si : H p-i-n diodes with an elevated defect density and thickness, the forward current exhibits an unusual transient property in which the onset of the forward current is delayed (Winbourne et al., 1988, Hack and Street, 1992). The delay time depends on the bias and the defect density, and can be milliseconds or longer. The effect arises from trapped space charge near the contacts, which inhibit the current flow. This is one of many examples in which device properties are affected by the amorphous nature of the material. Photovoltaic operation of the p-i-n diode occurs in slight forward bias, where power is generated by the illumination. The conversion efficiency is given by Efficiency = Voc I,, F F

(9-63)

The open circuit voltage, Voc, is related to (but less than) the Fermi energy difference in the n-type and p-type contacts, which in turn is related to the band gap energy and the width of the band tails, as discussed in Sec. 9.3 on electrical transport. The short circuit current, I,,, is given by the absorbed

586 104


I

I

lo4

I

106

1od

104

1od

Tu 10-7

(0

E

-f 0

3 E

a

w

a

109

iod 109

+-

9x10" x- 1.3 x 10" 0-1.8 x lo* v2x10"

10'0

10"

10-7

4,

I

I

I

1

I

0.2

0.4

0.6

0.8

1.0

1

Bias Vottage (V)

'"'0

m 0.2 0.4 0.6 0.8 1.0 Blas Voltage (V)

Figure 9-36. (left) measured forward J-V characteristics for a p-i-n diode in the annealed states and at various stages of current soahng; (right) numerical calculations of the J-V curves for defect densities corresponding to the measures conditions (Street and Hack, 1991).

light, and its maximum value occurs when every absorbed photon generates an electron-hole pair. In practice, there are losses due to absorption in and near the doped contacts which reduces the blue response. The fill factor, F,, describes the shape of the current-voltage characteristics, and is mainly determined by trapping at defect states. As the operating voltage approaches Voc, the internal field decreases and camer trapping becomes more pronounced. Hence the solar cell efficiency is sensitive to the defect density, and so is significantly affected by the metastable light induced defect generation which is described i n Sec. 9.4.6.1 The design of the single junction solar cell is a compromise between the need for a thick film to absorb as much of the solar

spectrum as possible and a thin film to collect the charge and maintain a high fill factor. Similarly, a large band gap gives a high value of V,, but reduces Zsc. The multijunction solar cell makes use of the flexibility offered by alloys of a-Si : H to improve the situation. A triple junction cell, for example, has three p-i-n diodes deposited directly on top of each other, each with a different band gap energy selected to absorb different parts of the solar spectrum. Alloys with germanium and carbon are typically used for these types of structure, and the approach has yielded cells with stabilized efficiencies of 13- 14% in a small area, limited primarily by the higher defect densities of the alloys compared to a-Si:H (Guha, 1996). Numerical modeling is important to


587

predict the device behavior of such complex solar cell structures, since analytical expressions give poor results. The modeling gives accurate results and confirms that the electronic properties of a-Si : H are indeed quite well understood. The reverse bias p-i-n photodiode is the operational mode of most light sensors, for which the photo-current, Zpc, is given by

The significant density of mid-gap states in a-Si : H ensures that the depletion layer is formed under conditions of deep depletion rather than inversion. The quasi-Fermi energy, EqF,describing the steady state occupancy of gap states is determined by the relative thermal excitation rates of electrons and holes, which are

(9-64)

(9-66)

IPC= VQE G e

where G is the absorbed photon flux. The quantum efficiency, vQE, of charge collection approaches unity when the defect density is low. The light detection sensitivity depends on the dark current, which is very low for p-i-n diodes. Figure 9-37 shows the typical time dependence of the dark current after a bias is applied (Street 1990). A depletion layer forms when the diode is in reverse bias, and the current is the sum of the transient release of the depletion charge, I d e p , (t), plus the steady-state current, Jss ]rev

( t )= Jdepl ( t )+ J S S

(9-67) A steady state is reached when these rates are equal, which occurs when E,, lies close to the middle of the band gap and is given

"J

E, + k T In CYI

where E, is the mobility gap, E,- E". The ratio of the transition rate prefactors, oEand u Hequals the ratio of the capture cross-sec-

(9-65)

3

(9-68)

I

'

Steady State 1V

lo-'

0

I

I

I

100

200

300

Time (sec)

Figure 9-37. Time dependence of the current after a reverse bias is applied, for different bias voltages. The steady state current at 1 V is indicated. The shaded region represents the charge depleted from the undoped layer (Street, 1990).


588

tions, and is roughly 10 (Street, 1984), so that EqFis held within about 0.05 eV of midgap in the depletion layer. Upon application of a bias voltage, depletion of the diode causes charges to be emitted between the equilibrium Fermi energy, E F , and EqF. Undoped a-Si: H is weakly n-type and therefore forms an electron depletion layer which extends from the p-i interface. The depletion charge, has been measured to be about 7 ~ 1 0cm-3 ' ~ in low-defect material (Qureshi et al., 1989), from which the density of mid-gap states is estimated to be 5 x 1 O I 5 cm3 eV-' (Street, 1993). Qureshi et a]. (1989) also show that the depletion charge is proportional to the dangling-bond density, confirming that these states are the dominant deep trap. The measurement provides an alternative method of measuring gap states, showing that device structures can provide fundamental materials measurements. The time dependence of the reverse current in Fig. 9-37 reflects the emission of electrons during formation of the depletion layer. The excess current decays with a time constant of order 100 s at room temperature, which is determined by the thermal release rate of carriers from states near mid-gap. The excess charge contained within the current transient is the depletion charge. A steady-state thermal generation current is reached when the rate of emission of holes to the valence band matches the emission of electrons to the conduction band. This establishes the trap quasi-Fermi energy as given by Eqs. (9-66) and (9-67), and the steady-state generation current is then (Street, 1990)

e,,,

JTG = N k T W E

where N is the density of states near midgap, (F)represents the weak dependence

:

on the electric field, and Vis the sample volume. Other possible contributions to the leakage current of a p-i-n sensor are injection through the contacts and, particularly in very small lithographically defined structures, leakage around the perimeter of the device (Schiff et al., 1996). Both contributions can be made small by careful optimization of the sensor and the edge passivation. A p-i-n diode of 1 pm thickness has a room temperature thermal generation current density of roughly lo-'' A cm-? at l V bias. The effective conductivity of lo-'' S2-' cm-' is several orders of magnitude lower than the bulk conductivity of undoped a-Si : H. The difference reflects the quality of the diode junctions and highlights the care that needs to be taken with the contacts in any conductivity measurement. The reverse bias thermal generation current given by Eq. (9-68) is proportional to the defect density and so provides a useful measurement of the defect properties of a-Si : H in the device. The example in Fig. 9-38 shows the defect density for currentinduced defects from measurements of the reverse current (Street, 1991). The defect density increases with inducing time, and the t1'3 kinetics obtained for light-induced defect creation agree with other measurement techniques described in Sec. 9.4.6.1. Defects are also generated by a forward current in the p-i-n diode by the same mechanism of electron-hole recombination. The rate of defect creation increases with the current and varies as approximately the square root of the time before saturation occurs. The different creation kinetics compared to light-induced defects arise because the densities of band-edge electrons and holes vary differently with illumination and forward current during defect creation. This technique for measuring the defect density also provided one of the first meas-


589

h

N

w

C

v)

w

io2

10’

1

IO

io2

io3

1o4

Inducing Time (min)

urements of induced defect recovery (Street, 1991).

9.5.3 Matrix Addressed Arrays TFT technology has enabled a successful manufacturing technology for large area active matrix arrays. Amorphous silicon is the leading technology for active matrix liquid crystal displays (AMLCDs), and active matrix image sensors incorporating p-i-n photodiodes are an emerging technology for medical X-ray imaging. The large area is essential for both display and sensor applications, since AMLCDs are progressing from lap-top computers to desk-top displays, and medical X-ray imagers of up to 35x43 cm in size are needed for general radiographic imaging. Matrix addressing provides a practical scheme to operate arrays with many pixels, and consists of intersecting address and data lines with a TFT at each intersection. Each address line activates the TITS along one column of the array and allows the transfer of signals between the pixels and the data lines, which are connected to the external electronics. After measurement of the signal is complete, the next address line is activated and the TFT holds the charge at the

pixel until the next addressing sequence. The array is addressed one line at a time and usually in sequence across the array, but in principle the order is arbitrary. The typical pixel structures of an AMLCD and an image sensor array are rather similar, and are illustrated in Fig. 9-39. The vertical address line is connected to the TFT gate, and the horizontal lines provide data addressing. Bias connections are also needed on the image sensor array, while the AMLCD has a backplane with the counter electrode. Additional pixel capacitance is sometimes added to either device. The TFTs hold the charge on the pixel when the TFT is off, and allow rapid charge transfer to or from the data lines when the T I T is addressed. In the AMLCD, a voltage is applied to the pixel, while in the image sensor, pixel charge is transferred to the external electronics. The demands on the TFT become more challenging as the performance of the array increases. The twisted nematic AMLCD structure is illustrated in Fig. 9-40. The liquid crystal film is about 5 pm thick and is sealed between the active matrix driver array and the backplane with the counter electrode and color filters. This assembly is placed between polarizers, with an illumination

590

9 Material Properties of Hydrogenated Amorphous Silicon Gate lines

GI I

G2

I

G3

I

I

J

t-n

-:--I

G4

I

K

Liquid crystal 1

I

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Sensor nixel

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Figure 9-40. Illustration of the structure of the twisted nematic active matrix liquid crystal display.

source behind the display. In the absence of a voltage on the pixel, the liquid crystal molecules are aligned parallel to the array substrate, with a 90” twist from one plate to the other. The liquid crystal is birefringent owing to its highly anisotropic molecular structure, and so the twist induces a rotation of the plane of polarization of light crossing the display. When a voltage is applied to the pixel. the dielectric anisotropy of the liquid crystal molecules causes them to line up perpendicular to the substrate, and rotation of the polarization is removed. The illumination is therefore switched with very little

Figure 9-39. Illustration of the pixel structure for active matrix LCD and image sensor arrays. The pixel capacitor is either the liquid crystal cell or the a-Si : H photodiode. TIT gates are addressed by a shift register and the data lines provide a voltage for the LCD or readout the charge from the image sensor.

power consumption and at sufficient speed for a video display (typical frequencies are 70 Hz to avoid flicker). At the time of writing, the AMLCD business has reached an annual market size of about $20 billion. Most of the displays are used in lap-top computers, but flat panel desk-top monitors are just being introduced, and liquid crystal displays are also quite widespread in video cameras. Color is required for virtually all the displays, which increases the pixel number by three or four times. However, an interesting opportunity exists for very high resolution monochrome displays for X-ray medical imaging. Some typical sizes and common formats for the large area applications are:

-

Lap-top displays: 10”- 14” (250-350 mm) 600 x 800 color pixels. Workstation displays: 14”-20” (350-500 mm) 1000x1280 color pixels. Medical imaging: > 17” (425 mm) 2000x2500 monochrome pixels. The manufacture of AMLCDs is now a maturing industry. The technology is derived from that for silicon integrated circuits and


relies on the ability for thin film deposition, photolithography, and etching on large glass substrates. Most arrays presently use Generation 3 process lines for which the substrate size is 55 x65 cm. Generation 4 process lines are presently in construction and will use an even larger substrate size of 80x80 cm or more. Turning now to the image sensors, each pixel senses the radiation that impinges on it and stores the corresponding charge until it is transferred to the external electronics (Street et al., 1990). The thin film p-i-n photodiode has a large enough capacitance to provide its own charge storage. Its sensitivity across the visible spectrum provides a sensor with full color imaging capability which is useful for document scanning. X-ray sensitivity for medical imaging is provided by two alternative approaches, as illustrated in Fig. 9-41. A phosphor screen, for example, made of Gd02S2:Tb, can be placed on the surface of the sensor array. Incident X-rays excite the phosphor which emits visible light which is detected by the image sensor. The light is collected very efficiently since the phosphor is in direct physical contact with the array. The alternative approach is to use a thick, X-ray sensitive photoconductor instead of the p-i-n diode, of which selenium is a good example (Lee et al., 1997). Both types of X-ray imaging system are in production.

-

incident x-ray photo-excitation ionization

591

The critical performance attributes of the image sensor are the spatial resolution, sensitivity, and dynamic range. For arrays that are typically 20-40 cm in size, the number of pixels ranges from 100 000 to 10 000 000, depending on the application. The spatial resolution is ultimately limited by the pixel size, and the interesting range for medical imaging is from 50 ym up to 300-500 ym, which is accessible by present a-Si : H technology. The signal developed at a pixel depends on the quantum efficiency of the sensor, and is also proportional to the sensor area in the pixel. Thus high resolution arrays are intrinsically less sensitive than low resolution arrays, in the sense that for a fixed illumination flux, the charge developed per pixel is smaller. The dynamic range is the ratio of the maximum signal to the noise, and values above 10 000 are obtained. Low noise performance is required and is provided by sensitive amplifiers attached to the array, as well as careful design and shielding of the system. The result is that Xray imagers can be made that approach the ideal performance when the noise limit is given by the shot noise of the incident Xray flux. The major application of large area image sensors is in medical imaging (Street and Antonuk, 1993; Street, 1999); although nondestructive testing is also important and the technique applies to a range of other X-

Incident x-ray photoexcitation x-ray photo-

conductor ionization scattered light e

---.-----

--_-____sensor

----------------

/capacito$ matrix-addressed readout circuitry

Figure 9-41. Comparison of i

I

!

X-ray detection using the phosphor/photodiode (left) and photo-conductor (right) methods.

592


ray applications. A-Si : H sensor arrays have a key technical advantage, because they address the fundamental problem that X-rays cannot be readily focused and so are most easily imaged with a large area detector. For example, a contact image of the large exposure area required for medical imaging is the most efficient acquisition technique for the phosphor approach shown in Fig. 9-41. Medical imaging is either performed in radiographic or fluoroscopic mode. A radiographic imager captures a single image in response to a brief X-ray exposure, while fluoroscopy captures a continuous real time sequence at rates of typically 30 frames per second. Radiographic images are particularly large and also require high resolution and high dynamic range. A size of 14”x17” (350x425 mm) is needed for chest imaging, with pixel size in the range 100-200 pm, so that the imagers must have about 10 million pixels and 3000-4000 lines. Figure 9-42 shows a portion of an image obtained by an array with 127 pm pixels, and illustrates the performance that can be obtained.

Figure 9-42. Example of a radiographic X-ray image acquired with an a-Si : H image sensor array.

9.6 Summary Much has been learned about hydrogenated amorphous silicon in the past two decades. Most of its interesting properties follow from the disorder of the atomic structure, which influences the material in several different ways. For example, the disorder inherent in the bond length and bond angle variations causes band tailing, strong carrier scattering and localization. The characteristic transport properties of a-Si : H follow, including the thermally activated electrical conduction and the dispersive carrier mobility. The random network structure also allows bonding configurations which are not allowed in the corresponding crystalline material. Coordination defects, in which a single atom has a different coordination from normal, are the elementary point defects in an amorphous semiconductor, of which the dangling bond is the primary example. The elementary defects in a crystalline material are the vacancy and interstitial, and the isolated dangling bond cannot occur, primarily because of the constraints of the long range order. Defect reactions and metastability are a general phenomenon in a-Si : H. Electronic excitations of all types induce structural changes, which are metastable at room temperature and anneal out above 100-150°C. The phenomena are broadly explained by the chemical bonding of the random network which allows a change of atomic coordination when an extra electron occupies a localized state. Consequently the equilibrium defect and dopant concentrations depend on the electron distribution and Fermi energy position. The hydrogen plays an important role in the metastability through its property of diffusing at moderate temperatures. Slow relaxation near the equilibrium temperature follows a stretched exponential time dependence, common to many disordered systems.

9.7 References

9.7 References Abraharns, E., Anderson, P. W., Licciardello, D. C., Ramakrishman, T. V. (1979), Phys. Rev. Lett. 42, 673. Allen, D. C., Joannopoulos, J. D. (1984), in: The Physics of Hydrogenated Amorphous Silicon II: Joannopoulos, J. D., Lucovsky, G. (Eds.). Heidelberg: Springer Verlag. Aljishi, S., Shu Jin, Ley, L., Wagner, S. (1990), Phys. Rev. Lett. 65, 629. Anderson, P. W. (1958), Phys. Rev. 109, 1492. Bar-Yam, Y., Joannopoulos, J. D. (1986), Phys. Rev. Lett. 56, 2203. Berner, H., Munz, P., Bucher, E., Kessler, F., Paasche, S . M. (1987),J. Non-Cryst. Solids97& 98, 847. Beyer, W., Overhof, H. (1984), Semiconductors and Semimetals, 21 C. Orlando: Academic Press, Chap. 8. Bohm, M., Blecher, F., Eckhardt, A., Schneider, B., Stertzel, J., Benthein, S., Keller, H., Lule, T., Rieve, P., Sommer, M. (1998), MRS Symp. Proc., 507,327. Branz, H. (1998), Solid State Commun. 105, 387. Carlson, D. E., Magee, C. W. (1978), Appl. Phys. Lett. 33,81. Carlson, D. E., Wronski, C. R. (1976), Appl. Phys. Lett. 28, 671. Ching, W. Y., Lam, D. J., Lin, C. C. (1980), Phys. Rev. B21, 2378. Chittick, R. C., Alexander, J. H., Sterling, H. E (1969), J. Electrochemical SOC.116, 77. Cutler, M., Mott, N. F. (1969), Phys. Rev. 181, 1336. Dersch, H., Stuke, J., Beichler, J. (1980), Appl. Phys. Lett. 38, 456. Ernin, D. (1977), Proc. 7th Int. Con& on Amorphous and Liquid Semiconductors: Spear, W. E. (Ed.). Edinburgh: CICL, 249. Etherington, G., Wright, A. C., Wenzel, J. T., Dove, J. C., Clarke, J. H., Sinclair, R. N. (1984), J. NonCryst. Solids 48, 265. Fedders, P. A., Carlssen, A. E. (1989), Phys. Rev. B. 39, 1134. Friedman, L. (1971), J. Non-Cryst. Solids 6, 329. Gallagher, A. (1986), Mat. Res. SOC.Symp. Proc. 70, 3. Guha, S. (1996), J. Non-Cryst. Solids 198-200, 1076. Hack, M., Street,R. A. (1992), J.App1. Phys. 72,2331. Hannay, N. B. (1967), Solid State Chemistry New Jersey: Prentice Hall. Hattori, K., Mori, T., Okarnoto, H., Harnakawa, Y. (1988), Phys. Rev. Lett. 60, 825. Jackson, W. B. (1982), Solid State Common. 44, 477. Jackson, W. B., Amer, N. M. (1982), Phys. Rev. B 25, 5559. Jackson, W. B., Kakalios, J. (1989), in: Amorphous Silicon and Related Materials Vol. 1: Fritzsche H. (Ed.) Singapore: World Scientific, 247.

593

Jackson, W. B., Biegelsen, D. K., Nemanich, R. J., Knights, J. C. (1983), Appl Phys. Lett. 42, 105. Jackson, W. B., Tsai, C. C., Thompson, R. (1990), Phys. Rev. Lett., 64, 56. Johnson,N. M., Herring, C., Chadi, D. J. (1986). Phys. Rev. Lett. 56, 769. Kakalios, J., Fritzsche, H. (1984), Phys. Rev. Lett. 53, 1602. Kakalios, J., Street, R. A,, Jackson, W. B. (1987), Phys. Rev. Lett. 59, 1037. Kimerling, L. (1978), Solid State Electronics 21, 1391. Knights, J. C., Lucovsky, G. (1980), CRC Critical Reviews in Solid State and Materials Sciences, 21, 21 1. Knights, J. C., Lujan, R. A. (1979), Appl. Phys. Lett. 35, 244. Kocka, J., Vanacek, M., Schauer, E (1987), J. NonCryst. Solids 97&98, 715. Kruangam, D., Deguchi, M., Hattori, Y., Toyama, T., Okamoto, H., Harnakawa, Y. (1987), Proc. MRS Syrnp. 95, 609. Kruhler, W., Pfeiderer, H., Plattner, R., Stetter, W. (1984), AIP Con5 Proc. 120, 31 1. Lang, D. V., Cohen, J. D., Harbison, J. P. (1982), Phys. Rev. B25, 5285. LeComber, P. G., Jones, D. I., Spear, W. E. (1977), Philos. Mag. 35, 1173. LeComber, P. G., Spear, W. E., Ghaith, A. (1979), Electronics Letters 15, 179. Lee, D. L., Cheung, L. K., Jeromin, L. S., Palecki, E. F., Rodericks, B. (1997), The Physics of Medical Imaging, Proc. SPIE 3032, 88. Ley, L., Richter, H., Karcher, R., Johnson, R. L., Reichardt, J. (1981), J. Phys. (Paris) C4, 753. Marshall, J. M., Street, R. A., Thompson, M. J. (1986), Philos. Mag. 54, 5 1. Martins, R., Fortunato, E. (1999), in: Technology and Applications of Amorphous Silicon, Street, R. A., (Ed.) Springer Verlag, Heidelberg. Meyer, W. von, Neldel, H. (1937), Z. Tech. Phys. 12, 588. Miki, H., Kawamoto, S., Horikawa, T., Maejirna, H., Sakamoto, H., Hayama, M., Onishi, Y. (1987), MRS Symp. Proc. 95. Boston: Materials Research Society, 43 1. Miyazaki, S., Ihara, Y.,Hirose, M. (1987), Phys. Rev. Lett. 59, 125. Mott, N. F. (1968), J. Non-Cryst. Solids I , 1. Mott, N. E (1969), Philos. Mag. 19, 835. Mott, N. F., Davis, E. A. (1979), Electronic Processes in Non-Crystalline Materials. Oxford: Oxford University Press. Mott, N. F., Kaveh, M. (1 985) Adv. Phys. 34 329. Nebel, C. E., Bauer, G. H., Gorn, M., Lechner, P. (1989), Proc. European Photovoltaic Con&, to be published. Okushi, H., Tokurnaru, Y., Yamasaki, S . , Oheda, H., Tanaka, K. (1982), Phys. Rev. B25,4313.

594


Orenstein, J., Kastner, M. (1981), Phys. Rev. Lett. 46, 1421. Pai, D. (1988), Proc. 4th Int. Con8 on Non Impact Technologies, SPSE, New Orleans, p. 20. Palma, F. (1999). in: Technology and Applications of Amorphous Silicon, Street, R. A. (Ed.), SpringerVerlag, Heidelberg. Pantelides, S. T. ( 1986), Phys. Rev. Lett. 57, 2979. Park, H. R., Liu, J. Z., Wagner, S. (1990), Appl. Phys. Lett., in press. Polk, D. E. (1971 ), J. Non-Cnst. Solids 5, 365. Powell, M. J., Glasse, C., Curran, J. E., Hughes, J. R., French, I.D., Martin,B. F. (1998).MRSSymp. Proc. 507, 91. Qureshi, S., Perez-Mendez, V., Kaplan, S. N., Fujieda, I., Cho, G., Street, R. A. (1989), MRS Syrnp. Proc. 149, 649. Santos.P., Hundhausen, M., Ley, L. (1986),Phys.Rev. B3.3, 1516. Santos, P., Johnson. N. M., Street, R. A , , (1991), Phys. Rev. Lett. 67, 2682. Scher, H., Montroll, E. W. (1975), Phys. Rev. B 1 2 , 2455. Schiff. E. A.. Street, R. A , , Weisfield, R . L. (1996), J. Non-Cnst. Solids 198-200, 1155. Schulke, W. (1981), Philos. Mag. B43, 451. Shockley, W., Moll, J. L. (1960), Phys. Rev. 119, 1480. Smith, Z. E., Wagner, S. 11987), Phys. Rei: Lett. 59, 688. Snell, A . J., LeComber, P. G., Mackenzie, K. D., Spear, W. E., Doghmane, A . (1983), J . Non-Cnsr. Solids 59&60, 1187. Spear, W. E., LeComber. P. G. (1975), Solid Stare Common. 17, 1 193. Staebler, D. L., Wronski, C. R. (1977), Appl. Phys. Lett. 31, 292. Street, R. A. (1982), Phys. Rev. Lett. 49, 1187. Street, R. A. ( 1 9 8 4 ~Philos. Mag. 8 4 9 , L15. Street, R. A. (1987). Proc. SPIESynp. 763, IO. Street, R. A. (1990), Appl. Phys. Lett. 57, 1334. Street, R. A. (1991 ), Appl. Phys. Lett. 59, 1084. Street, R. A. (1993), J. Non-Cpst. Solids 164-166, 643. Street, R. A. (1999), in: Technology and Applications ofAmorphous Silicon, Street, R. A. (Ed.), Springer Verlag, Heidelberg. Street, R. A., Antonuk, L. E. (1993), IEEE Circuits and Devices, Vol. 9, No. 4. 38. Street, R. A , , Biegelsen, D. K. (l982), Solid State Comm. 44, 501. Street, R. A., Hack, M. (1991),MRSSyrnp. Proc. 219, 135. Street, R . A., Winer, K. (19891, Phys. Rev. B40, 6263. Street, R. A., Biegelsen, D.K., Jackson, W. B., Johnson, K. M., Stutzmann, M. (1985a), Philos. Mag. B52, 235. Street, R. A., Thompson, M. J.. Johnson, N. M. (1985b), Philos. Mag. B51, I .

Street, R. A., Kakalios, J., Tsai, C. C., Hayes, T. M. (1987a), Phys. Rev. B35, 1316. Street, R. A,, Tsai, C. C., Kakalios, J., Jackson, W. B. (1987b), Philos. Mag. B56, 305. Street, R. A.,Hack,M., Jackson, W.B. (1988a),Phys. Rev. B37, 4209. Street, R. A., Kakalios, J., Hack, M. (1988b), Phys. Rev. B38, 5603. Street, R. A., Nelson, S . , Antonuk, L. E., Perez Mendez, V. ( 1 9 9 0 ~MRS Symp. Proc. 192, 441. Street, R. A., Apte, R. B., Ready, S. E., Weisfield, R. L., Nylen, P. (1998), MRS Symp. Proc. 487, 399. Stutzmann, M. (1987), Philos. Mug. B56, 63. Stutzmann, M. (1989), PhilosMag. B60, 531. Stutzmann, M., Street, R. A. (1985), Phys. Rev. Lett. 54, 1836. Stutzmann, M . , Jackson, W. B., Tsai, C. C. (1985), Phys. Rev, B32, 23. Stutzmann, M., Jackson, W. B., Tsai, C. C. (1986), Phvs. Rev. B34, 63. Stutzmann, M., Biegelsen, D.K., Street, R. A. (1987), Phys. Rev. B35, 5666. Stutzmann, M., Street, R. A., Tsai, C. C., Boyce, J. B., Ready, S. E. (1989), J . Appl. Phys. 66, 569. Swartz, G. A. (1984), Appl. Phys. Lett. 44, 697. Tanaka, K., Matsuda, A. (1987), Mater. Sci. Reports 3, 142. Tanielian, M. (1982), Philos. Mag. B45,435. Thompson, M. J., Tuan, H. C. (1986), IEDMTech. Digest. Los Angeles: IEDM, 192. Thompson, M., Johnson, N. M., Nemanich, R. J.,Tsai, C. C. (1981), Appl. Phys. Lett. 39, 274. Tiedje, T., Rose, A. (1980). Solid State Commun. 37, 49. Tiedje, T., Cebulka, J. M., Morel, D. L., Abeles, B. (1981), Phvs. Rev. Lett. 46, 1425. Tsukada, T. (1999), in: Technology and Applications of Amorphous Silicon, Street, R. A. (Ed.), Springer Verlag, Heidelberg. Weaire, D.(1971), Phys. Rev. Lett. 26, 1541. Winbourne, G., Xu, L., Silver, M., (1988), MRSSymp. Proc 118. 501. Winer, K. (1989), Phys. Rev. Lett. 63, 1487. Winer, K. (1990), Phys. Rev. B41, 7952. Winer, K., Ley, L. (1987), Phys. Rev. B36, 6072. Winer, K., Street, R. A. (1989), Phys. Rev. Lett. 63, 880. Winer, K., Wooten, E (1984), Phys. stut. sol. ( b ) 124, 473. Winer, K., Hirabayashi, I., Ley, L. (1988), Phys. Rev. B38, 7680. Wronski, C. R., Lee, S., Hicks, M., Kumar, S. (1989), Phys. Re),. Lett. 63, 1420. Yang, L., Abeles, B., Eberhardt, W., Stasiewski, H., Sondericker, D.(1987), Phys. Rev. B35, 9395. Zafar, S., Schiff, E. A. (1989), Phys. Rev. B40, in press.

9.7 References

General Reading Elliott, S. R. (1990), Physics of Amorphous Materials. New York: Longman. Joannoupoulos, J. D., Lucovsky, D. (Eds.) (1984), The Physics of Hydrogenated Amorphous Silicon I and IZ. Berlin: Springer-Verlag. Mott, N. E, Davis, E. A. (1979), Electronic Processes in Non-Crystalline Materials. Oxford: Oxford University Press.

595

Pankove, J. (Ed.) (1984), Semiconductors and Semimetals Vol. 2 1: Hydrogenated Amorphous Silica. Orlando: Academic Press. Street, R. A. (1991), Hydrogenated Amorphous Silicon. Cambridge: Cambridge University Press. Zallen, R. (1983), The Physics of Amorphous Solids, New York: Wiley.

10 High-Temperature Properties of Transition Elements in Silicon Wolfgang Schroter. Michael Seibt IV. Physikalisches Institut der Georg-August-Universitat Gottingen. Germany

Dieter Gilles Wacker Siltronic AG. Burghausen. Germany 599 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Transition Elements in Intrinsic Silicon . . . . . . . . . . . . . . . . . 604 10.2 10.2.1 Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 609 10.2.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Solubility and Diffusion in Extrinsic Silicon . . . . . . . . . . . . . . 615 615 10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 618 10.3.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 High Temperature Electronic Structure . . . . . . . . . . . . . . . . . . 619 10.4 Precipitation of Transition Elements in Silicon . . . . . . . . . . . . . 621 10.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 621 622 10.4.1.1 Chemical Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . 624 10.4.1.2 Precipitate Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 625 10.4.1.3 Spatial Distribution of Precipitates . . . . . . . . . . . . . . . . . . . . 10.4.2 Atomic Structure of Silicide Precipitates . . . . . . . . . . . . . . . . . 625 10.4.2.1 Precipitation Without Volume Change: Nickel and Cobalt . . . . . . . . 625 10.4.2.2 Precipitation with Volume Expansion: Copper and Palladium . . . . . . 630 633 10.4.3 Heterogeneous Precipitation . . . . . . . . . . . . . . . . . . . . . . . . 633 10.4.3.1 Iron in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 10.4.3.2 Copper in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 10.4.3.3 Nickel in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Electrical Properties of Silicide Precipitates . . . . . . . . . . . . . . . . 635 10.4.4.1 Recombination Behavior of Silicide Precipitates . . . . . . . . . . . . . 636 10.4.4.2 Spectroscopy of Deep States at Silicide Precipitates . . . . . . . . . . . 636 10.4.5 Silicide Precipitation at Si/SiO, Interfaces . . . . . . . . . . . . . . . . 638 10.5 Gettering Techniques and Mechanisms . . . . . . . . . . . . . . . . . 639 Introduction to Gettering Mechanisms . . . . . . . . . . . . . . . . . . . 639 10.5.1 10.5.2 Internal Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 10.5.2.1 Oxygen Precipitation Gettering . . . . . . . . . . . . . . . . . . . . . . 640 10.5.2.2 p/p+ Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 646 10.5.3 External Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

598

10.5.3.1 10.5.3.2 10.5.3.3 10.5.3.4 10.5.3.5 10.6 10.7

10 High-Temperature Properties of Transition Elements in Silicon

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poly-Silicon Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . Aluminum Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavity Gettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phosphorus Diffusion Gettering, Segregation and Injection Gettering . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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DLTS EBIC erfc f.c.c. FZ HRTEM H-site ic

601

melting point of silicon annealing temperature temperature at which a the solution of a metallic impurity M becomes supersatured temperature -time profile during cooling to room temperature elastic contribution to the migration enthalpy elastic energy of metal atom on tetrahedral (T) or hexagonal (H) interstitial site vacancy solubility of metal atoms M in atomic fractions solubility of metal atoms M in silicon in atomic fractions atomic fraction of metal atoms M in the Si-M melt coordination number number of nearest neighbors of tetrahedral (T) or hexagonal (H) interstitial site central force constant; number of silicon self-interstitials emitted per precipitating metal atom normalized derivative of elastic modulus K with respect to temperature T ; number of vacancies emitted per precipitating metal atom static dielectric constant of silicon permittivity of free space activity coefficient of M in the Si-M melt internal equilibration time of electronic states at extended defects cooling rate distribution coefficient chemical potential of metal atoms M in the reference phase chemical potential of metal atoms M in silicon chemical potential of metal atoms M in the intermetallic compound cm3) atomic volume of silicon (= 2 density-of-states of extended defects charge state of metal atom time constant of precipitation time constant of pairing reaction aluminum gettering body-centered cubic Czochral ski deep level transient spectroscopy electron beam induced current error function complement face-centered cubic float zone high-resolution transmission electron microscopy hexagonal interstitial site intermetallic phase

602 MOS NAA

PDG PSG SF SIMS TEM T-site

10 High-Temperalure Properties of Transition Elements in Silicon

metal-oxide-semiconductor neutron activation analysis phosphorus diffusion gettering phosphorus silicate glass stacking fault secondary ion mass spectroscopy transmission electron microscopy tetrahedral interstitial site

10.1 Introduction

10.1 Introduction The transition elements form a class of impurities in silicon, which combine unusual physical properties with very strong effects on devices. Both aspects have initiated intensive research activities, especially on the 3d-row from Sc to Cu, Pd and Au and Pt. Recent efforts to develop low-cost silicon materials for solar cell applications have revived activities focused on precipitation and gettering in silicon materials which contain dislocations, grain boundaries, or oxygen precipitates. Due to extensive experimental and theoretical investigations, the origin and background of multiple deep levels introduced into the band gap of silicon by transition elements are now well understood. The theoretical principles underlying this electronic structure are described in Chap. 4 of this Volume. In this chapter, an outline will be given of what is known about solubility and diffusion (Sect. 10.2), the electronic structure at high temperature (Sec. 10.3), precipitation (Sec. 10.4), and gettering (Sec. 10.5) of transition elements. Experimental data and models describing these properties already give a consistent overall picture of this class of impurities. They also elucidate the role that the transition elements play in device fabrication, as described by Graff (1995). Gettering techniques, dealt with in Sect. 10.5, have been developed to remove transition elements from the device active area. Our main concern in this section is to model gettering mechanisms and to specify the experimental conditions under which these mechanisms operate. Extremely small quantities of transition elements are sufficient to produce malfunctions of devices. It is reported in the literature that a concentration of IO" Ti atoms/ cm3 cannot be tolerated in a solar cell (Chen

603

et al., 1979). The ubiquitous presence of transition elements such as Fe, Ni and Cu along with high diffusivities, makes them highly undesirable in device processes. This is one reason why clean room facilities are imperative in device manufacturing. Up to now, effective removal of metallic impurities during or after any device step to regions outside of the device-active area, termed gettering, has been implemented to ensure a sufficient yield of devices. Some transition elements like Au and Pt have also been of beneficial use in device technology in cases where short lifetimes are desirable, as for some switching devices. The diamond lattice is arather open structure. Its large interstices provide the same space for atoms on substitutional and tetrahedral interstitial sites (T sites). Both sites also have the same symmetry ( T d ) . Metallic atoms usually prefer one of these sites since they have to adjust their electronic configuration to the bond structure of the host. The 3d transition elements in intrinsic silicon prefer the T site, whereas most of the 4d and 5d transition elements are predominantly dissolved on substitutional sites. However, the partial solution enthalpies AH$i) are unusually large (1.5-2.1 eV, see Sec. 10.2.1) and the maximum solubilities are extremely small (about 0.2 ppm for Cr and about 20 ppm for Ni). The major part of this enthalpy must be of electronic origin, since strain energy can only account for a small fraction of AH$".On the other hand, the migration enthalpy of the 3d elements, decreasing from about 1.8 eV for Ti to about 0.4 eV for Co, seems to be mainly of elastic origin and can be accounted for using a simple hard-sphere model (Sec. 10.2.2). A slight shift of the Fermi level from its position in intrinsic silicon toward the conduction band creates a new situation for 3d elements: the solubility of the substitutional species becomes comparable to that of the

604


interstitial species. It strongly increases further with the Fermi level approaching the conduction band, the effective diffusivity of the metallic atoms decreasing to the same extent (see Sec. 10.3). The solubility enhancement is caused by acceptor levels of the substitutional species, whose occupation contributes to the partial enthalpy of solution. Aposition of the Fermi energy slightly above mid gap marks the transition from the interstitial to the substitutional species determining the behavior of Mn, Fe, Co, and

cu.

Solubility together with diffusion data measured in extrinsic silicon can be used to study the electronic structure of metallic atoms at high temperatures. It was found that the low temperature defect configuration of interstitial Mn, Fe, and Co becomes unstable above 1000 K, which is possibly analogous to a point defect-droplet transition of the silicon self-interstitial, as conjectured by Seeger and Chik (1968) (Sec. 10.3). Precipitation from supersaturated solid solutions of transition elements in silicon proceeds with a large gain of free energy of the order of 1 eV per metal atom, an unknown situation in metallic systems. Descriptions of precipitation phenomena in terms of classical nucleation theory no longer seem to be appropriate. Instead, particle compositions or morphologies are selected which establish fast precipitation growth and hence fast relaxation of the supersaturation. Current understanding of this topic is summarized in Sec. 10.4. The initial precipitation stages of Ni and Cu, which are obtained by fast quenching from the diffusion temperature, are thin Nisi, and Cu,Si platelets. As evidenced by deep level transient spectroscopy, these platelets are extended multi-electron defects without detectable decoration by other impurities and are therefore ideal objects to investigate the electronic properties of extended defects. To remove

transition elements from device-active parts of silicon wafers, annealing procedures have been developed by which segregation or precipitation is induced in predetermined parts (gettering, see Sec. 10.5). Three basically different mechanisms are discussed. Gettering may occur by precipitation at extended defects (oxide particles, dislocations, stacking faults, etc,) generated by a preceding treatment from a supersaturated solution (relaxation gettering). It may also result from the diffusion of metallic atoms into parts of the wafer, whose thermodynamic properties have been modified (segregation gettering). Finally, it may come about by a flux of self-interstitials towards a sink, which induces a flux and an accumulation of solute atoms close to the sink (injection gettering).

10.2 Transition Elements in Intrinsic Silicon 10.2.1 Solubility

In 1983, solubility data of 3d transition elements in Si were published by Weber (1983), covering the transition elements from Cr to Cu within the 3d row of the periodic table. His results are consistent with independent experiments for Mn (Gilles et al., 1986), Fe (Isobe et al., 1989), Co (Utzig and Gilles, 1989), and Ni (Aalberts and Verheijke, 1962; Yoshida and Furusho, 1964). For Cu, Weber’s data are not in agreement with those of Dorward and Kirkaldy (1968). In addition, solubility data for Ti (Chen et al., 1979; Hocine and Mathiot, 1988) and V (Sadoh and Nakashima, 199 1) are available. For Sc, there have not been any systematic studies yet, but the solubility appears to be comparable to that of Ti (Lemke, 1981). Pd is the only element within the 4d row for which solubility data are available

10.2 Transition Elements in Intrinsic Silicon

(Frank, 1991). For Pt, Lisiak and Milnes (1 975) published NAA data which are consistent with those of Hauber et al. (1989) obtained above the eutectic temperature. For Au, consistent data are available from Collins et al. (1957), Trumbore (1960), Sprokel and Fairfield (1 9 6 3 , Dorward and Kirkaldy (1968), and Stolwijk et al. (1984). As can be seen in Figs. 10-1 and 10-2, the solubilities of the 3d elements, the 4d element Pd, and the 5d elements Pt and Au show a strong decrease with decreasing temperature. Compared to group I11 (B, AI, Ga) or group V (P, As, Sb) impurities, the maximum solubility is quite small. For Cu and Ni, up to about 10" atoms/cm3 are soluble,

T in OC

40012001000 800

600

Ui

1.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 4

T in OC

1001200 1000 800

"\ \

600

501

\

co

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1. 1000/T in K-'

Figure 10.1. Solubilities [MI,, ( T )of 3d elements in silicon. Curves are shown for the temperature ranges covered by experimental data. References are for Cu: solid line (Dorward and Kirkaldy, 1968), dashed line (Weber, 1983); Ni: solid line [common fit to neutron activation analysis (NAA) data of Weber (1983), Aalberts and Verheijke (1962) and radio-tracer data of Yoshida and Furusho (1964)], dashed line (Weber, 1983); Ti: solid line (Hocine and Mathiot, 1988, 1989), dashed line (Chen et al., 1979), for other references, see Tables 10-1 and 10-2.

nnnlr

in K-1

Figure 10-2. Solubilities [MI,, ( T )of the 4d element Pd and the 5d elements Pt and Au in silicon (data for Ni and Cu are shown for comparison); for references, see Tables 10-1 and 10-2; for Pt, the line represents the data of Lisiak and Milnes (1975), which are consistent with data obtained by Hauber (1986) above T,", .

for Pd, Pt, and Au about 1017~ r n and - ~ for Ti, V, Cr, Mn, Fe, and Co it is less than 4 x 1O I 6 atoms/cm3. These values can be compared with a maximum solubility of about 1021atoms/cm3 for phosphorus and about 5 x lo2' atoms/cm3 for boron in silicon. On the silicon-rich side of the phase diagram, some systems Si-M have a eutectic temperature well below the melting point of silicon (Ti!)=1687 K). They show retrograde solubility, i.e., the temperature of maximum solubility is higher than the eutectic temperature T,,, (Mn, Fe, Co, Ni, Cu, Pd, Pt, and Au). The solubility of M in Si depends on enthalpies and entropies. Therefore it has to be measured with the external phase that is given by the phase diagram as being in equilib-

606


rium with the solid solution of M in Si (Dorward and Kirkaldy, 1968). When starting with a pure metal on top of silicon, one has to make certain that the correct phase is formed and that it is equilibrated with the solid solution at the annealing temperature. Solubilities of transition elements are defined by the equilibrium between silicon and the Si-richest silicide M,Si,. if T e T,,, , and between silicon and the liquid solution, if T > T,,, . The Si-richest silicides are of type MSiz for the 3d transition elements from Ti to Ni, Cu,Si for Cu, and PtSi for Pt. In the case of Pd, the Si-richest silicide (PdSi) decomposes into Pd,Si and Si at 824 “C, which is slightly below the eutectic temperature of 892 “C. As will be shown in Sec. 10.3, for Mn, Fe, and Co, and presumably also for Ti and Cr, the dominant species at high temperatures is Mio’. The measured values of QM and the prefactor can then be given a clear thermodynamic meaning, which follows from equal chemical potentials of the two phases with respect to M p:’

( rp ) = p E i )(?: pt

XM,eq)

(10-1)

where the index ic denotes the intermetallic phase and p E c )and pEi)are the chemical potentials (partial free enthalpies per atom) of M in the intermetallic phase and M:” in Si, respectively. In the formulas, the concentration is expressed in atomic fractions XM eq = [M],,/Ni = [MIo’],, lN, with Ni = 5 x lo’* being the density of tetrahedral interstitial sites in silicon. Describing the deviations from the ideal solution of M!’’ in Si by an excess partial free enthalpy of mixing gives AGdi(/Si) = AHg/Si)- T

( 10-2)

which leads to &iJ (T,

&’(x

p ) is the reference state of where metal in solid solution and is chosen here to be the intermetallic phase, AG$Isi) is the free enthalpy required to transfer one metal atom from the intermetallic phase to an interstitial site in silicon as a neutral species M!’), AHg”si)is the partial heat of mixing, which is zero for the ideal solution, and is the excess partial entropy of mixing. AH:'^'' results from a variation of the phonon frequencies in mixing M into Si. Another approach derives these mixing quantities from the data measured for T > Teu,, i.e., from the equilibrium between M in solid silicon and in liquid solution. The procedure is slightly more complicated, since the liquid solution is not ideal and therefore numerical fitting of the liquidus line must be carried out. However, the advantage of this procedure is its reliance on experimental data; the liquid boundary phase should have the composition corresponding to the liquidus line at the annealing temperature, since diffusion in liquids is usually fast. Equilibrium between the solid solution of the metallic impurity M in Si [atomic fraction and of M in the SiM melt [atomic fraction x$,,,] is given by the solidus line and defines the partial excess enthalpy A HEa)and entropy AS^'"''

XC;%+

(1 0-4)

x M is the distribution coefficient and y u is the activity coefficient of M in the Si-M melt. Since differences between the specific heats of the solid phase and of the melt are small, they may be neglected, giving A Hgi/l)= A HZ)- A H(si) M

(10-3) (idsi ) +k~Tln(x,,eq)+AGM

P, XM,eq)=&)(Z

p)

and A SgilL) = AS,(0- A s c i )

( 10-5)


607

where AH$ and AS$ are the enthalpy and (ic: intermetallic compound). However, dientropy of fusion of M, respectively, and rect evidence that the reaction between silAH$i) and AS$') are the partial excess enicon and the metallic film on top of it has thalpy and entropy of M in the solid Si-M really led to the formation of the equilibrisolution with respect to the pure metals, reum silicide phase is missing in these soluspecti vel y. bility experiments. Table 10-1 summarizes the solution paExceptions are the experiments of Dorrameters, obtained for TcT,,, from an ward and Kirkaldy (1968), who equilibratArrhenius plot of measured data. The preed silicon with solid Cu3Si via the vapor ~ ~ (see Table 10-1), and of Utzig and factor has been expressed as 5 ~ 1 0 phase exp (S,lk,) cm-3 and S, has been listed. Gilles (1989) who proved equilibrium with QM and S, have the import of thermodyCoSi, at the sample surfaces by means of namic quantities, if the equilibrium silicide MoBbauer spectroscopy. The latter data are phase according to the phase diagram has also consistent with those of Weber (1983). In all other cases, we can only perform a formed and has been brought into equilibrium with the Si-M solid solution during anconsistency check between the results obnealing. This then gives QM=AH$ISi) and tained from T > T,,, with the second analySM -- A s g l s i ) where AHgIsi) and ASgIsi) sis, and those obtained for Tc T,,, with the first analysis. The partial solution enthalpare the partial excess enthalpy and entropy ies and entropies should be the same if the of solution, respectively, of M in Si with the external solid phase is the correct one. For equilibrium silicide as the reference phase

Table 10-1. Solubility data of 3d transition elements in silicon.

Ti V Cr Mn

Fe

co Ni

cu Pd Pt

3.0 3.05 4.04 2.79 2.81 2.78 2.94 2.83 1.68 1.49 1.7520.02 I .64 2.43

3.9 4.2 11 4.8 7.3 6.9 8.2 7.6 3.2 2.4 4.9k0.3 1.2 7.0

1000- I250 950- 1330 959- 1200 900- 1335 900- 1142 920- 1078 900- 1206 700- 1260 500-993 500-802 650-802 600-892 800-950

1330 1400 1335 1I42 1206 1259 993 802 892 979

Chen et al. (1979) Hocine and Mathiot (1988) Sadoh and Nakashima (1991) Weber (1983) Weber (1983) Gilles et al. (1986) Weber (1983) Weber (1983) Weber (1 983) Weber (1983) Dorward and Kirkaldy (1968) Frank (1991)' Lisiak and Milnes (1975)'

1.39 1.26 1.28 1.02 0.6 1 1.07 0.91 0.04 1.05 1.24

1.61 1.66 2.78 1.51 1.79 1.78 2.33 1.76 0.76 1.45 1.71 0.59 1.19

AH:': partial enthalpy of formation of the metal silicide in equilibrium with M in solid silicon below T,,, [data taken from Murarka (1983) and Schlesinger (1990)l; the value for iron is taken from Krentsis et al. (1963); AHE": partial enthalpy of solution of the transition element in silicon; 'the values of Q , and SMdetermined by Schroter and Seibt (1999a); formation enthalpies of compounds are usually given in kJ/mol, where 1 mole of M,Si, is the sum of the atomic weights of the constituents (also called "g-atom" or "mol of atoms"). Values of AHi',-are needed in electronvolts per metal atom and are obtained by multiplying the value given in kJ/mol by (x+y)196.4 x. a

608


y$)= I , Eq. (10-4) describes an ideal solution of M in liquid Si. This approximation has been shown to be justified for Si-Cu and has been applied to this system by Dorward and Kirkaldy (1968). It has also been applied to Mn and Ni in Si by Weber. The nonideality of the solution gives rise to corrections of about 0.2-0.3 eV for Mn, Fe, and Ni in Si (see Table 10-2), but it should be noted that the experimental data are reproduced by both descriptions. Available results for the 3d elements Mn, Fe, Ni, and Cu, the 4d element Pd as well as the 5d elements Pt and Au have been summarized in Table 10-2. In addition, enthalpies and entropies of fusion of the pure met-

als are given, which allow the calculation of HEi)and Sgi)using Eq. (10-5);these results can be compared to the respective values obtained from measurements below T,,, listed in Table 10-1. Agreement of the two values shows that the correct boundary conditions were established during the experiment at T C T,,,; this is the case for Cu in the experiment by Dorward and Kirkaldy (1968), for Mn and Fe, but not for Ni and Cu in the experiment by Weber ( I 983). However, at present the measurement and analysis of xM(T) still have two drawbacks: (1) the liquidus line and the parameters describing xM( T ) are rather sensitive to the temperature of fusion of sil-

Table 10-2. Solubility parameters for manganese. iron, nickel, copper, palladium, platinum, and gold obtained from an analysis of x M ( T l a . Element

AH (eV) (:Al=1)

Mn Fe Ni

cu

Pd Pt Au

1.71 (1.92) 2.15 12.40) 1.04 ( 1.25 ) 1.62 i 1.60) 1.36 ( 1.34) 2.43 ( 1.46 j 2.49 (1.60) 1.98 (1.92)

’

AHA

(ks) I y $ = 1)

(eV)

0.82 (1.18) 3.4 (4.1) -0.6 (-0.17) 3.6 (3.4) 0.6 (-0.8) 12.0 (-0.1) 12.2 (0.82) 3.8 (2.8)

0.14 0.15 0.18 0.14 0. I4 0.23

1.2 1.2 1.1 1.3

0.13

1.2

AS $“In

ASA (k,i

AHG” (eVi

ASE” (kB)

1.1

1.85 2.30 1.22 1.76 1.50 2.66 2.72 2.1 1

1.9 4.4 0.6 4.8 1.7 13.3 13.5 5.0

1.o

Reference

b c, d

b e

c. f

c.g c. h c.

I

To calculate the solubility of M in silicon at a given temperature T > T,,,, insert the parameters obtained for yh= I [given as ( ) i n columns 1 and 21 into Eq. (IO-4), which gives x M and xglb, taking x $ ) , : , ~from the phase diagram: x $ is obtained via .x$ = 1 -3-ki from the liquidus line, which for an ideal solution is determined by the fusion parameters of silicon (taken by Weber as Tl,= 1685 K, AHki=O.525 eV, AS&=3.62 k , ) (Weber, 1983). Actually, the liquidus lines of liquid Si-M are nonideal. For the silicon-rich side of the phase diagram, the deviations of the activity coefficient [y$i=~x(Sd)’/x~i] from one are described by relative partial enthalpies and excess entropies. which are proportional to (.&,j2 and are given by AH$i =OS2 ( x h ) ’ eV and A,S&=3.62 kB (&)* k, for copper i n silicon, by AH&=0.94 (x;)’ eV and A,Sg1=5.12 kB (&)’kg for palladium, and by AHsi= 1.41 (I$)’ eV. AeS,*,=8.68k , ( . ~ $ ) ~fork the ~ 3d elements from titanium to nickel. Assuming that these relations are valid at all temperatures and compositions, using the Gibbs-Duhem equation for x $ the same quantities AH,*; and AeS,*lwith xkl replacing x $ on the right sides are obtained. This leads to y h = e x p ( A H h l k , T-A,S&IkB); fit obtained by Schroter and Seibt (1999a); experimental data (eight values for T>T,,,) obtained by Young (1982); e experimental data and analysis by Dorward and Kirkaldy (1968); experimental data published by Frank (1991): experimental data after Hauber (1986) and Hauber et al. (1989); experimental data of Lisiak and Milnes (1975); expenmental data of Collins et al. (1957) Trumbore (1960). Sprokel and Fairfield (19651, Dorward and Kirkaldy (1968), and Stolwijk et al. (1984). a


icon Ti?(Weber, 1983), such that its value has to be taken as a free parameter in the analysis of data with unknown error of T registration, (2) the formation enthalpies of metal silicides are not known with sufficient precision (Schlesinger, 1990). 10.2.2 Diffusion

Compilations of diffusion data (Weber, 1983; Graff, 1986; Weber and Gilles, 1990; Schroter and Seibt, 1999a) show that 3d transition elements and the 4d element Pd are among the fastest diffusing impurities in intrinsic silicon with diffusion coefficients as high as to lo4 cm2/s (Co, Ni, and Cu at IOOO'C, see Table 10-4). This explains why these impurities are harmful during device processing. For instance, for a wafer with a thickness of typically 500 pm, penetration might take less than 10 s for metal impurities. High diffusion coefficients along with low migration enthalpies are characteristics of simple interstitial diffusion, in which case no diffusion vehicle is needed and the formation of covalent bonds with silicon neighbors does not take place. For a simple interstitial diffusion mechanism, an Arrhenius-type behavior of the diffusion coefficient is expected

2 7

D = @ e x p --

( 10-6)

where Do is the pre-exponential factor, and A H g " the migration barrier. Electron paramagnetic resonance measurements have confirmed that Cr, Mn and Fe, which can be quenched to room temperature, predominantly occupy the tetrahedral interstitial site (Ludwig and Woodbury, 1962). In addition, the electronic structure of the dominant defect, as measured by deep-level transient spectroscopy [for review, see Graff (1986) and Schroter and Seibt (1999b)], is in agreement with total

609

energy calculations of the interstitial metal atom [Beeler et al. (1985), Zunger (1986), see Chap. 4 of this Volume]. There have been reports about a substitutional component of Ni (Kitagawa and Nakashima, 1987, 1989), but it should be emphasized that its concentration is several orders of magnitude smaller than the total Ni solubility. For Mn and Cr, a substitutional component has been found after co-diffusion with Cu (Ludwig and Woodbury, 1962). The mechanism of M, formation upon Cu precipitation is not yet understood. As can be seen from Fig. 10-3, there is a clear distinction between the diffusivities of impurities that occupy substitutional sites (e.g., shallow dopants) and those on interstitial sites. However, there is also a considerable variation in the diffusivities within the 3d row. From Ti to Ni the diffusivity at 1000°C increases by about six orders of magnitude and the migration enthalpy decreases by a factor of four. Surprisingly, a rather simple hard sphere model, which takes into account the variation of atomic size within the 3d row, satisfactorily explains these trends, which have been discussed as a "diffusion puzzle" for a long time (Zunger, 1986). In this model, Utzig (1 989) calculates the elastic energy required to move Mi from one tetrahedral interstitial site (T site) to the next via the hexagonal interstitial site (H site) as the saddle point (see Fig. 10-4). The pre-exponential factor is calculated as the migration entropy ASL"'' using an approach by Zener (1952). The T site is in the center of the tetrahedron formed by four Si atoms [ z ( ~ ) =41. The distance between the center of the tetrahedron and its comers is the same as that between two neighboring silicon lattice sites, do.The hard sphere radius of the T site is then = do- rsi, rsi = d0/2 being the atomic radius of silicon, so that R(=)=0.5 do= 1.I7 A (0.1 17 nm). The H site is in the center of a

61 0

10 High-Temperalure Properties of Transition Elements in Silicon

T in OC

s E

-e 0

u)

E 0

10.131,

,

,

,

,

,

,

,

,

I

0.6 0.7 0.8 0.9 1 .O 1.1 1.2 1 .3 1.4 1 .5 1OOOn in K-l

Figure 10-3. High temperature diffusion coefficients of the 3d elements and Pd in silicon. There is a clear distinction between the diffusivities of impurities that occupy substitutional sites and those of the metal impurities that occupy tetrahedral interstitial sites. There is also a considerable variation of D within the 3d row, which has been explained by a simple hard-sphere model (see text): references are for Ti: Hocine and Mathiot ( 1988) (solid line), Nakashima and Hashimoto (1992) (dashed line): V: Nakashima and Hashimoto (1992); Cr: Bendik et al. (1970) (solid line), Zhu et al. (1989) (dashed line); Mn: Gilles et al. (1986): Fe: Struthers (1956) (solid line), Isobe et al. (1989) (dashed line); Co: Utzig and Gilles (1989); Ni: Bakhadyrkhanov et al. (1980); Cu: Hall and Racette (1964);Pd: Frank (1991 ). The values for Cu have been measured in highly boron doped silicon (see text).

hexagonal ring formed by six Si atoms [ ~ ' ~ ) =and 6 ]has a hard sphere radius R ( H ) = 0.95 do -rsi=0.45 do= 1.05 A (0.105 nm). Inserting an atom of radius rM into the interstitial site requires an elastic energy given by (T.H)]? ,(T.H) if rM ,R(T,H) c

else

Figure 10-4. The tetrahedral (T-site: 0 ) and the hexagonal interstitial site (H-site: +) in the diamond lattice. 3d-elements occupy the T-site and diffuse presumably via the H-site as the saddle point.

where a is the central force constant per next ) the T site and for Si-neighbor [ z ( ~ for the H size] : a =3.02 eV/A (Keating, 1966). The difference A Ue,= U:y)(rM)-ULT)(rM) is a contribution to AH$"'). The atomic radii of the 3d elements, taken as the distance of closest approach in the metal (see Table 10-3; Hall, 1967), have been corrected following an empirical rule detected by Goldschmidt (1928). Depending on the coordination number z , the interatomic distance was found to be about 3% less if z is 8 instead of 12, 4% less if it is 6 (H site), and 12% less if it is 4 (T site). In Table 10-3 calculated values of AUe, are compared with measured values of AHLmi' for 3d elements. The agreement is quite satisfactory. The sharp drop of AHhmi' from Ti to Cr, and the subsequent slow decrease, are quite well reproduced by this simple model. Note that the estimation of

61 1


AU,, for copper is for CU,!'), while the experimental data are for CUI+). Under the condition that the diffusion barrier is primarily due to elastic strain, which appears to be fulfilled for the 3d elements, Zener (1952) has given an approximate formula to calculate the migration entropy

( 10-7)

where K is the elastic modulus and Ti? the melting point of silicon. Taking Do= 1/6 d i f e x p [AShmi)],wherefis the Debye frequency, measured values of the pre-exponential factor of diffusion Do can also be compared with the prediction of the simple hard sphere model (see Table 10-3), again showing remarkable agreement.

Table 10-3. Comparison of experimental migration barriers A H F ) with calculated elastic contributions AUeIa. rMb

(nm) Ti V Cr

0.1467 0.1338 0.1357 0.1267 Mn 0.1306 0.1261 Fe 0.1260 Co 0.1252 Ni 0.1244 CU' 0.1276

AU,, AH?) (ev) (ev) 2.15 0.99 1.15 0.50 0.75 0.46 0.46 0.42 0.38 0.55

1.79 1.55 0.81 0.70 0.68 0.37 0.47

0.18

Do(AUel) (cm2/s)

DO (cm2/s)

0.52x10-* 1 . 4 5 ~ 1 0 - * ~ . I O X ~ O 9- ~. 0 ~ 1 0 - ~ 0.23xlO-* l.OxlO-* 0.14x10-* 1 . 7 0 ~ 1 0 - 0~ . 6 9 ~ 1 0 - ~ 1.34~10-~ 1 . 3 4 ~ 1 0 - ~1 . 3 ~ 1 0 - ~ 1 . 3 0 ~ 1 0 - ~0 . 9 0 ~ 1 0 - ~ 1 . 2 5 ~ 1 0 - ~2.0X10-3 1 4 . 5 ~ 1 0 - ~0 . 3 ~ 1 0 - ~

~

Utzig (1989); the references from which Utzig has taken experimental values can be found in Table 10-4 [compare rM is the atomic radius of the respective metal atom (Darken and Guroy, 1953); ' copper is known to diffuse as Cu,'+)in silicon, while atomic radii are for neutral atoms (see text). a

Mr)];

Diffusion coefficients D, estimated from the time necessary to saturate Si samples with a 3d element M (Weber, 1983) showed that care must be taken if D, is evaluated from aconcentration profile. One type of experiment to measure the diffusion coefficient D, is usually performed with either a finite source or an infinite source of M at the surface of the silicon. Finite source experiments avoiding silicide formation have been found to suffer from silicon surface reactions which are difficult to control. The alternative boundary condition is a constant surface concentration, realized by the deposition of a metal film onto the silicon surface and by inducing the formation of the silicide phase. There is an easy check of the basic requirement in this experiment. For a well-defined boundary condition, the surface concentration, extrapolated from the diffusion profile, must coincide with the solubility data at the diffusion temperature. Let us consider as an example the diffusion of Mn in Si. In the temperature range between 900 "C and 1200 "C symmetric diffusion profiles were obtained after in-diffusion from opposite surfaces of Si samples (Fig. 10-5a; Gilles et al., 1986). Surfaceconcentrations were found to be in good agreement with the solubility data (Fig. 10-5b), which means that boundary conditions independent of time have been achieved. Such checks are indispensible to accurately determine migration enthalpies below 1 eV. Note that this agreement does not imply that the equilibrium phase has formed, as discussed in Sec. 10.2.1. An independent method for measuring D M below 150°C is the result of a study of the pairing reaction MI+'+A',-' (Mi As), where A, is a shallow acceptor. The kinetics of pair formation is determined by the diffusion of M!+). According to Reiss et al. (1956) the diffusing species Mf+' is captured by Ai-) via electrostatic attraction as soon as

-


612

1200

IO

(a)

-05

C

05

T in “C 1030

1100

930

10

Xla

Figure 10-5. Solubility and diffusion coefficient of Mn in silicon (after Gilles et al., 1986). (a) Concentration profiles determined by the tracer method after diffusion from both surfaces into specimens of thickness 2 a for different temperatures: 1200°C (O), 1080°C (m), 990°C (o), 920°C (A). (b) Temperature dependence of diffusion coefficients, D, calculated from diffusion profiles of the total Mn-concentration (tracer method, solid square, cp. (a))or those of interstitial Mn (DLTS, open squares). The surface concentration of those profiles (tracer method, solid circles) agree well with the saturation concentrations.

the electrostatic energy exceeds the thermal energy, k,T, of the ion. The identity of these two energy terms defines the capture radius, R , such that from the relaxation time constant, tp, the diffusivity, D:’), of MIC’ can be calculated

where E = 11.7 is the static dielectric constant of silicon, the permittivity of free space and, e the unity of electric charge. In order to determine the relaxation time, isolated Mi has to be quenched to a temperature below which the ratio [M!” A:-’]/ [MI”] > 1 , and at which the relaxation time is experimentally accessible. While for

Cr, Fe, and Mn both conditions can be fulfilled below 100°C, for Co, relaxation is already complete during or immediately after quenching (Bergholz, 1983). The 1/T range is significantly larger for the low T ( [MI,,, and 2. metal atoms have to be sufficiently mobile. [MI

Figure 10-11. Free energy curves of silicon (F")and of a metal silicide (F")for the calculation of the gain of chemical free energy Afchem due to precipitate formation. Note that the diagram only schematically reproduces the true situation since [MI, and [MI,, are less than 100 ppm and [MI,, is of the order of 30%.

whose composition [MI' is illustrated by parallel tangents to Fsi and Ficat [MI, and [MI', respectively. Considering the change in free energy due to decomposition of the supersaturated solution shows that the gain of chemical free energy per metal atom Afchemcan be visualized by the vertical distance of the tangent to Fsi and Ficat the composition [M]=[M]'[e.g., Haasen(1978)l. To estimate Afchem, it is assumed that the composition of the precipitate is that of the silicide (rather than [MI'). Bearing in mind that concentrations of transition elements in silicon are well below 100 ppm, a good approximation of Afchem is given by

in which uM ([MI) is the activity of metal atoms in silicon for a given concentration [MI, and [MI,, is the solubility of metal atoms in equilibrium with the silicide. Within this model for a regular solution, the activ-

Figure 10-12a shows the temperature dependence of Afchem for solid solutions of B, Cu, 0, and Fe in Si which have been equilibrated at T = 1200°C. In order to appreciate the unqiue properties of the fast diffusing 3d transition metals, it is instructive to compare the behaviors of 0 and Cu in Si. According to Fig. 10-12a, these two impurities show almost identical chemical driving forces for precipitation due to their similar solubilities in Si. In order to nucleate SiO, precipitates, annealing times of the order of 1 h at 700°C are applied in standard annealing cycles for internal gettering (Shimura, 1988). For diffusion-limited precipitation, Fig. 10-12b shows that the relevant time scale for copper precipitation differs by at least nine orders of magnitude, Le., annealing times of several microseconds at 700 "C lead to similar results for Cu as 1 h for 0 in Si. In addition, at temperatures typical for the growth of SiO, precipitates (=1000 "C), the diffusion coefficient of 0 in Si is still seven orders of magnitude below that of copper at 700°C. From these considerations we may conclude that fast diffusing transition metals in Si are likely to precipitate during or after cooling from high temperatures, as will be described in Sec. 10.4.2. Finally, it should be mentioned that the different behavior of Fe in Si (Sec. 10.4.3.1), which is the heaviest 3d transition element that can be quenched on interstitial sites, may sim-

624


Table 10-8. Composition M,Si, of impurity precipitates in silicon and related volume changes.

YV A

$

50

0)

40

c. 2 2

lmpurity

Silicide

a-pa

AVIVob (’lo)

30

0)

-

20

0

10

C ’5

Fe

L

co Ni

0

a)

temperature (K)

cu

Pd 0

a - FeSil

-0.11

-5.5

p-FeSi, CoSi, Nisi, Cu3Si Pd2Si SiOz

-0.08 -0.025 =OS 0.55 0.50

-4.0 -1.125 = 150 110 100

a a (/3) denotes the number of self-interstitials (vacancies) emitted per precipitated impurity atom for strainfree precipitates; AV/Vo denotes the relative volume change [AV/Vo=x/y (a-P)I.

.-

U

temperature (K)

b)

Figure 10-12. a ) Chemical driving force for precipitation of impurities in silicon as a function of temperature: the impurity concentration corresponds to the respective solubility at 1200°C [solubility data taken from Weber (1983) (Fe, Cu). Mikkelsen (1986) (O), and Nobili (1988) (B)]. b) Diffusion coefficients of the same impurities as a function of temperature [diffusion data from Weber (1983) (Fe), lstrato\, et al. (1998a) (Cu), Mikkelsen (1986) ( 0 )and Brown et al. (1988) (B)].

ply be due to its considerably smaller mobility compared to Cu.

10.4.1.2 Precipitate Composition With the exception of Au in Si (Baumann and Schroter, 1991), precipitates with compositions of those silicides i n equilibrium with the Si have been observed. In Table 10-8. the data for interstitially dissolved metal impurities are summarized together with the data for 0 i n Si. For Fe, Co, and Ni the precipitating phase is the disilicide (MSi2), whereas metal-rich silicides (M,Si x 2 2) form for Cu and Pd. An important con-

sequence is the volume change associated with precipitation, which results in elastic strains unless strain relaxation via plastic deformation or the production (annihilation) of intrinsic point defects occurs. To elucidate the role of intrinsic point defects, we may describe the formation of a M,Si, precipitate from Mi by the quasi-chemical reaction (Marioton and Gosele, 1988) x Mi+y Si+P,

t P,+, + x

a I + x P V (10-20)

where I and V are the silicon self-interstitial and vacancy, respectively, and P, is a precipitate containing n metal atoms; a (p)denotes the number of I (V) emitted per precipitated metal atom. An upper limit of (a-p) can be calculated assuming strain-free precipitates (see Table 10-8), in which case the volume change AVcanbeexpressedasAV=x (a-p) Qsi, where Qsi is the atomic volume of silicon ( Qsi=2 . l 0-23cm3). Table 10-8 further provides the relative volume change given by AV/V,=x(y (a-p). It is easily seen from Table 10-8 that Co and Ni precipitate almost without volume change, whereas a considerable volume expansion is associated with the formation of Cu3Si and PdzSi precipitates leading to I emission in amounts com-

10.4 Precipitation of Transition Elements in Silicon

parable to that associated with SiO, precipitate formation in silicon. For Fe, a small volume contraction occurs indicating that V emission or I absorption relieves misfit strains.

10.4.1.3 Spatial Distribution of Precipitates Besides the size and shape, the spatial distribution of silicide precipitates critically depends on experimental parameters like the impurity concentration and the cooling conditions from high temperatures. For defect-free Si precipitation during cooling first occurs at the wafer surface leading to a concentration gradient of dissolved impurity atoms towards the surface. As a result, a high density of precipitates is observed directly at wafer surfaces, whereas a precipitate-free zone (not to be mixed up with the denuded zone discussed for internal gettering in Sec. 10.5) forms in the regions below the surface. The width of this zone varies from a few micrometers for fast quenching to almost the total wafer thickness for slow cooling. This phenomenon is the underlying process of haze formation, which was recognized in the 1960s (Pomerantz, 1967). Subsequently, the detection of haze has been developed as a semi-quantitative tool to evaluate the efficiency of gettering techniques (Graff, 1983; Graff et al., 1985; Falster and Bergholz, 1990; Graff, 1995).

10.4.2 Atomic Structure of Silicide Precipitates Before we describe the experimental results, we shall briefly discuss how experiments suitable for the investigation of fast diffusing transition metal precipitates could be designed. Transmission electron microscopy (TEM) is so far the only experimental technique available for structural analysis.

625

From our preceding description of haze formation (Se. 10.4.1.3), it is clear that utmost care has to be taken as to which part of the Si samples the TEM foils are prepared from. Hence the use of cross-section geometry is highly favored. Since the macroscopic spatial distribution as well as precipitate size and composition are critically dependent on the cooling rate after in-diffusion of the transition element under investigation, this parameter is of utmost importance and should be specified in any study of the precipitation of fast-diffusing impurities as well as the initial metal impurity concentration. Recently, it has been recognized for copper in silicon that the type of dopant may also affect the precipitation behavior (see Sec. 10.4.2.2 and 10.4.4). The next two sections describe results obtained in n-type FZ-Si unless otherwise stated. Section 10.4.2.1 summarizes results obtained for cobalt and nickel in silicon, which precipitate without a considerable volume change. It will be seen that silicide particles with a metastable structure form upon quenching; these structurally transform into energetically more favorable configurations without long range diffusion upon annealing at low temperatures. Precipitation with volume expansion is described in Sec. 10.4.2.2, where observations for copper in silicon are described in detail. Here plateshaped precipitates surrounded by extrinsic stacking faults are observed in samples quenched from high temperatures; these transform into precipitate colonies during further annealing.

10.4.2.1 Precipitation Without Volume Change: Nickel and Cobalt For cobalt and nickel, the silicides in equilibrium with silicon are CoSi, and Nisi,, respectively. They have the face centered cubic CaF, structure with lattice pa-

626


rameters differing from that of silicon by less than 1.2%, so that the specific volume of the silicon atoms is nearly equal for both silicon and the silicides (see Table 10-8). Hence coherency strains are only important in the case of large precipitates. Furthermore, both silicides grow epitaxially on Si, indicating small interfacial energies (for a detailed description of the epitaxial growth of these silicides on silicon, see Chap. 8 of this Volume). All these properties along with the high mobility of interstitial cobalt and nickel imply that precipitate nucleation and growth can occur at small supersaturation and lead to low energy configurations. However, as we shall see, both 3d impurities form metastable precipitates upon quenching from high temperatures. Using high resolution TEM (HRTEM), Seibt and Schroter (1 989) investigated the early stages of nickel precipitation in silicon. In their experiments a nickel film was evaporated onto one surface of the samples before annealing at temperatures between 850°C and 1050°C. The in-diffusion was terminated by quenching the samples in ethylene glycol leading to cooling rates of about 500-1000 Ws. Figure 10-13 is a lattice image of a typical plate-shaped precipitate obtained in the bulkof the samples, Le., at least 100 pm away from any surface. These platelets are parallel to Si { 11 1 ] planes. Whereas their diameters vary between 0.4 and 0.9 pm for the above quenching conditions, platelets as small as 7 nm have been observed after annealing at 750 "C and quenching into 10%NaOH from a vertical furnace (Riedel, 1995; Riedel et al., 1995). Analysis of lattice images revealed that the precipitates consist of only two { 1 1 1 ) N i s i z planes (Fig. 10-13b). This implies that each nickel atom in these platelets belongs to the precipitate/matrix interface. The atomic structure of this interface could

Figure 10-13. a) High resolution electron micrograph of an Nisi, platelet obtained after rapid quenching from 1050°C ( f = l O O O Ws, n-type FZ-Si); b) magnified detail showing the relative shift of the Si lattice above and below the platelet (Seibt and Schroter, 1989).

be determined by means of micrographs like Fig. 10-13 b, leading to the balls-and-sticks model shown in Fig. 10-14. Both { 111) interfaces are built up by Si-Si bonds leaving nickel atoms in sevenfold coordination (compared to eight silicon nearest neighbors in bulk Nisi,). Hence, since no nickel atom in these precipitates has cubic surroundings, as would be the case for particle shapes associated with a low energy, these structures are metastable. The interfacial structure described above agrees with that obtained from Nisi, films epitaxially grown on Si ( 1 1 1 ) substrates (Cherns et al. 1982; also see Chapt. 8 of this Volume). For geometrical reasons, platelets with the observed interfacial atomic arrangement introduce a


627

a silicon atom by a nickel atom pushing the silicon atom into the antibonding site (nearest tetrahedral interstitial site), as shown in Fig. 10-15 a. This process is essentially different from that of interstitial diffusion, 1.02nrn where atoms jump from one interstitial site to another. Hence it might be speculated that I the process depicted in Fig. 10-15a involves a high barrier, which does not allow fast precipitate growth. In Fig. 10-15b a possible core structure of the bounding a/4 (1 11) dislocation is shown, indicating that the disloFigure 10-14. Ball and stick model of Nisi, platelets cation provides distorted sites where reobserved after quenching; the precipitate/matrix interfaces are built up by Si-Si bonds; the arrow indicates placement of a silicon atom by a nickel atthe rigid shift of the Si lattice above and below the om can occur more easily. Accepting that platelet (after Seibt and Schroter, 1989) the bounding dislocations establish a high incorporation rate of nickel atoms into the platelets, relaxation of the supersaturation displacement in the silicon lattice, as is intakes place faster for platelets containing dicated by the arrow in Fig. 10-14. The disonly two { 1 11 ] Nisi2layers compared to alplacement has to be compensated for by a ternative structures, since they maximize dislocation bounding the precipitate (boundthe length of the bounding dislocation for a ing dislocation). These dislocations, which given precipitate volume. In the case of cobalt precipitation in silihave Burgers vectors of type a/4 (1 11) incon, similar behavior can be expected since clined with respect to the platelet normal, CoSi, and Nisi, are isomorphous and the have been predicted on the basis of crystaldiffusivities are comparable (see Fig. 10-3); lographic considerations (Pond, 1985) and Mossbauer spectroscopy (Utzig, 1988) was have been observed using conventional used to separate different cobalt species in TEM (Seibt and Schroter, 1989). silicon. In low-doped p- and n-type samples, Owing to the strain energy of the boundall the cobalt atoms were found on noncuing dislocation and the energy associated with the large area of the NiSi2/(lll)Si interfaces, the shape of the precipitates described above must be viewed as energetically unfavorable. It has been proposed that this structure is selected for kinetic reasons. Si It can grow much faster compared to more compact particles, because the bounding dislocation is the most efficient channel for II (b) (a) the incorporation of nickel atoms into the Figure 10-15. (a) Basic process of Nisi, formation precipitates (Seibt and Schroter, 1989). For from Si: replacement of a Si atom by a nickel atom illustration, consider the transformation pushing the silicon atom to the tetrahedral interstitial from the diamond structure (silicon) to the site; (b) possible core structure of the bounding fluoride structure (Nisi2). Basically, this a/4(111) dislocation providing sites for easy incorporation of Ni atoms. process can be viewed as the replacement of

1

m


628

bic sites directly after quenching. In complete analogy to the case of nickel i n silicon, this species is considered as cobalt atoms in the precipitate/matrix interface of the platelets, implying that thin platelets are also formed in this case. On additional annealing at medium temperatures (200-600 "C), a transformation of cobalt atoms from noncubic to cubic sites is observed due to an increase of the platelet thickness. Furthermore, it could be shown that the fraction fcublc of cobalt atoms on cubic sites only depends on the annealing temperature (Fig. 10-16a), and a constant value offcubicis reached quite rapidly (e.g., 4 min at 400°C: Fig. 10-16b). Both these rw

I

V

0

". 200

T in "C

,

(b)

600

4m

(a)

-

observations led to the conclusion that the transformation process occurs without long range diffusion, i.e., it is due to rearrangement of atoms within individual precipitates in order to establish energetically more favourable configurations. This process of internal ripening has been observed directly by TEM for nickel in silicon. Annealing at 320°C of samples containing Nisi, platelets consisting of two ( 11 1 } planes results in cylindrical precipitates (Fig. 10-17) while leaving the precipitate density unchanged. Structural investigations at various stages of internal ripening show that the transformation proceeds by island formation at the border of the platelets and their subsequent growth (Riedel, 1995). The driving force for internal ripening is mainly the reduction of strain energy associated with the bounding dislocation and the decreasing surface-tovolume ratio which reduces the interface energy. The same driving forces led to conventional Ostwald ripening (see Haasen, 1978) during annealing at higher temperatures. For the case of Nisi, platelets it has been shown (Seibt and Schroter, 1989) that during additional annealing in the temperature range of 500-900°C the precipitate density

., 10

100

loo0

t in min

Figure 10-16. (a) Fraction fcublc of cobalt atoms on cubic lattice sites as a function of annealing temperature; (b) fcub,c as a function of annealing time for T=400°C (data after Utzig, 1988); open and closed circles refer to Cz and FZ silicon, respectively.

Figure 10-17. Nisi, platelet after internal ripening at 320°C for 30 min; the HRTEM micrograph shows the platelet to be of type B, Le., in a twin orientation with respect to the silicon matrix.


is reduced by a factor of 10-100 (depending on the annealing temperature). The platelet thickness increases by the same factor so that the precipitate diameter remains almost unaffected. This leads to a decrease in the interfacial energy per precipitated nickel atom by at least one order of magnitude. This kind of particle ripening is accompanied by a process shown in Fig. 10-18 in a series of electron micrographs. The platelets now contain regions of different orientation relationships of Nisi, and silicon. In both types, Nisi, shares a (1 11) direction with the silicon and is either aligned with silicon (type A) or rotated about this (1 11) axis by 180" (type B), i.e., it is twinned. Type A and B orientations are frequently observed in epitaxial Nisi, films grown on { 1 1 1 ) substrates [Cherns et al. (1 982), Tung et al. (1 983), see also Chap. 8 of this Volume]. Figure 10-18a-c shows type A regions at the border of a Nisi, platelet and a central type B grain. A change in the platelet thickness is often observed at A-B boundaries (Fig. 10-18d). In contrast to epitaxially grown films, where small initial nickel coverages lead to type B formation (Tung et al., 1983), no correlation is observed between the platelet thickness and the type of orientation. Instead, the fraction of type B grains was found to depend on the annealing temperature and was interpreted as due to an A to B transformation occurring at temperatures below 800°C (Seibt and Schroter, 1989). However, the reason for this behavior has not yet been clarified. Hitherto, we have not considered the influence of elastic strains due to lattice mismatch between the matrix and the precipitate. This is only justified for small precipitates. Under conditions of slow cooling (cooling rates e 50 Ws), however, platelets with thicknesses of up to 100 nm and diameters of up to 100 pm have been observed

629

Figure 10-18. (a) Bright-field TEM micrograph showing an inclined Nisi, platelet after quenching and annealing at 500°C; (b) and (c) show type B and type A regions of the platelet, respectively; (d) lattice image of an A-B boundary associated with a thickness change (Seibt and Schroter, 1989).

in near surface regions of silicon wafers in the case of nickel (Stacy et al., 1981; Augustus, 1983a, b; Picker and Dobson, 1972; Seibt and Graff, 1988a, b; Cerva and Wendt, 1989a; Kola et al., 1989). Another precipitate shape is also observed, i.e., modified pyramidal shapes (octahedral, tetrahedral) which always exhibit type A orientation. Figure 10-19 shows an example of such an Nisi, precipitate, which is heavily deformed, as indicated by the dislocations within the particle. Part of the misfit strain has also relaxed by producing dislocations in the silicon matrix, which often form closed loops (see arrows in Fig. 10-19).

630


Figure 10-19. Heavily deformed polyhedral Nisi, precipitate of type A orientation, and type B oriented platelet (P) after slow cooling ( T = 5 Us) from 1050°C; part of the lattice mismatch has relaxed by the production of dislocations in the Si matrix (arrows).

10.4.2.2 Precipitation with Volume Expansion: Copper and Palladium Investigation of the precipitation behavior of copper in silicon started as early as 1956, when Dash used copper to decorate dislocations in silicon crystals. Since then, numerous investigations have been performed, mostly stimulated by the technological importance of copper as a common contamination in silicon device production, not least due to the recent introduction of copper interconnects in silicon integrated circuit technology. From TEM studies, it is well known that copper precipitates in the form of star-shaped colonies if medium or slow cooling rates are applied [an extensive list of references is given in Seibt and Graff ( 1988 a)]. The colonies consist of small copper silicide particles forming planar arrangements parallel to Si { 1 IO} or Si { 00 1 } planes (Nes and Washburn, 1971), which are

bounded by edge-type dislocation loops. Figure 10-20a is a low magnification electron micrograph of such a star-shaped agglomeration. A magnified section showing the bounding dislocations is given in Fig. 10-20b. In the late 1970s a model was developed by Nes and co-workers (Nes and Washburn, 1971; Nes, 1974; Solberg and Nes, 1978a) which describes the colony growth. The model is based on earlier observations of NbC precipitation at stacking faults in niobium-containing austenitic stainless steels (Silcock and Tunstall, 1964). Before we briefly describe the basic features of this model, we have to consider the composition of the small precipitates. Early suggestions that the particles consist of b.c.c. CuSi (Nes and Lunde, 1972; El Kajbaji and Thibault, 1995) or f.c.c. CuSi (Das, 1972) have been disproved on the basis of electron diffraction studies (Solberg, 1978; Seibt and Graff, 1988a). Extensive investigations using TEM led to the proposal that the precipitates have the f’-Cu,Si structure (Solberg, 1978), which is the low temperature modification of the phase in equilibrium with silicon below the eutectic temperature (see Sec. 10.2.1). Recent analysis of data obtained from direct lattice imaging (Seibt et al., 1998a) are consistent with a hexagonal structure ( a = 0.708 nm and c = 0.738 nm) closely related to the f-Cu,Si structure described by Solberg (1978). Since the exact crystal structure of the precipitating copper silicide is of minor importance for the subsequent discussion, we shall refer to it as Cu,Si. Its formation in the silicon matrix is associated with a volume expansion of about 150%, which can be relaxed by the emission of one silicon selfinterstitial (I) per two precipitating copper atoms, or by the absorption of one vacancy (V). For simplicity, we shall discuss the processes involved in terms of I emission only, and neglect contributions from V.


631

Within the model of Nes and co-workers, the first stage of precipitation is the nucleation of particles at a dislocation (Fig. 10-21a). Subsequent growth of the Cu3Si precipitates leads to the emission of I, which force the dislocation to climb (Fig. 10-21 b) by the absorption of I. During this process, the precipitates are dragged by the climbing dislocation, which is indicated in Fig. 10-21 b by the dashed line showing the original position of the dislocation. Particle dragging has been observed by means of in situ TEM for Cu3Si particles (Solberg and Nes, 1978 b). As the Cu$i precipitates grow, their mobility decreases and dislocation segments between them bow out until dislocation unpinning occurs (Fig. 10-21c). Now the situation of the first stage of precipitation is restored and new particles can nucleate at the dislocation. This mode of precipitation has been termed repeated nucleation and growth on climbing dislocations, which is an autocatalytic process. The model describes the formation of star-shaped colonies once dislocations are present. However, modern silicon materials are virtually dislocation free and, for them at least, the model does not account for the early stages of precipitation. The question

41

4

Figure 10-20. a) Low magnification TEM micrograph of a Cu particle colony obtained in the surface

4

632


of how particle colonies nucleate has been answered by means of HRTEM (Seibt, 1990). The investigation of bulk regions of copperdiffused samples quenched from high temperatures (cooling rate 1000 W s ) revealed plate shaped defects parallel to Si [ 1 1 1 ) planes. Figure 10-22a is an HRTEM micrograph showing a central platelet (P) with a thickness of 3 nm, which is surrounded by an extrinsic stacking fault. Figure 10-22b shows small Cu,Si particles which have nucleated at the stacking fault (arrows).

Still earlier stages of copper precipitation show defect configurations similar to that shown in Fig. 10-22a, with stacking fault diameters of 100 nm and platelets with thicknesses of about 1 nm (Seibt et a]., 1998a). Annealing at temperatures below 400°C leads to internal ripening (see Sec. 10.4.2.l ) , resulting in spherical silicide particles in the center of stacking faults instead of platelets. The result of the usual Ostwald ripening is the formation of silicide particle colonies (Seibt et al., 1998a) like those obtained after slow cooling. The process leading to colony formation has been modeled on the basis of these observations, and is schematically shown in Fig. 10-23 a-d. The homogeneous nucleation of a precipitate is shown in Fig. 10-23 a. For small particle sizes the volume expansion is expected to be compensated for by elastic strains, leading to plate-shaped precipitates (Fig. 10-23b), which is the shape realizing minimum strain energy (Nabarro, 1940).

(C)

Figure 10-22. a) Cu silicide platelet (P)associated with a stacking fault (SF) after quenching from 1050°C ( i = l O O O W s ) :b) lattice image of a stacking fault and of spherical Cu silicide precipitates (arrows) (Seibt, 1990).

(d)

Figure 10-23. Early stages of colony formation (after Seibt, 1990): (a) homogeneous nucleation of a Cu silicide precipitate; (b) growth leads to plate-shaped particles in order to minimize the strain energy; (c) further growth initiates theemission of Si,, which condensate into stacking-fault-like defects; (d) nucleation of spherical Cu silicide precipitates at the stackingfault-like defect; this configuration may be viewed as a particle colony in the embryonic stage.


Further growth of the platelets initiates strain relaxation via I emission, which finally leads to nucleation of bounding extrinsic stacking faults (Fig. 10-23c). These in turn can act as nucleation sites for further Cu3Si particles, resulting in configurations that can be viewed as particle colonies in the embryonic stage (Fig. 10-23d). During subsequent growth, the stacking faults may transform into perfect loops producing the colonies described above, or stay stable, since they are sometimes observed as parts of large colonies. The process described above is very similar to the precipitation of oxygen in silicon (see Chap. 5 of this Volume). However, one basic difference is due to the fact that oxygen in silicon has a much lower mobility than silicon self-interstitials, whereas copper diffusion is at least two orders of magnitude faster than I diffusion. Unless sinks are present for I, this implies that the copper precipitation kinetics are limited by the mobility of I (Marioton and Gosele, 1988). Hence the formation of I sinks via stacking fault nucleation (Fig. 10-23c) is a pre-requisite for fast relaxation of the copper supersaturation. Recent experiments (Flink et al., 1999b) indicate that copper precipitation is charge-controlled (see Sec. 10.4.4), with no evidence of self-interstitials to limit the precipitation kinetics. This corroborates the conclusion that sinks for self-interstitials should form very early in the precipitation process of copper in silicon. Finally, we briefly want to mention that copper silicide precipitates of tetrahedral shape have been observed at Si { 001 } surfaces after in-diffusion of copper using rapid thermal annealing (Kola et al., 1989). These authors argue that the free surface acts as a sink for I, so that dislocations or stacking faults are not required to realize fast precipitation.

633

10.4.3 Heterogeneous Precipitation In the preceding sections, metal impurity precipitation in silicon was discussed for FZ materials that are dislocation-free and do not contain microdefects related to oxygen precipitation in silicon. From the point of view of relaxation-induced gettering processes (see Sec. 10.5.2) and of multicrystalline silicon materials used for solar cell applications, however, precipitation at defects such as dislocations, grain boundaries, or microdefects is of importance. In the following sections, the results on heterogeneous metal impurity precipitation are described. Rather than giving a complete summary of the numerous experimental works on this subject, an attempt is made to focus on the mechanisms involved and their relation to the homogeneous precipitation behavior described in Sec. 10.4.2 onwards.

10.4.3.1 Iron in Silicon For iron, the silicides in equilibrium with silicon are a -FeSi, and P-FeSi, above and below 915 "C, respectively. The tetragonal crystal structure of the metallic a-FeSi, and the orthorhombic structure of the semiconducting P-FeSi, are closely related to the CaF, structure of nickel and cobalt disilicides (Dusausoy et al., 1971). Table 10-8 shows that the formation of precipitates of either structure is associated with a volume contraction, Le., vacancy emission and selfinterstitial absorption may relieve misfit strains. Iron precipitation under most experimental conditions seems to be heterogeneous in nature, Le., it is always assisted by some extended defects. In particular, the simultaneous presence of fast precipitating (haze forming) metal impurities enhances the iron precipitation rate [see Graff (19931. Early

634


studies have been performed under conditions of silicon device fabrication (Cullis and Katz, 1974: Augustus et al., 1980).They show rod-like a-FeSi, precipitates with the rodaxis parallel to Si(l10)directions.These are directions of small misfit (0.9%) and allow coherent a-FeSi,/( l l l 1 Si interfaces. Figure 10-24 shows a TEM micrograph of such an a-FeSi, precipitate after iron diffusion at 1100°C and cooling (Seibt and Graff, 1989). Besides strain contrast, a small spherical precipitate attached to the tip of the particle is observed. The latter is most probably due to residual copper contamination and emphasizes the heterogeneous nature of iron precipitation. It is interesting to note that iron and copper precipitation are associated with volume changes of opposite sign (see Table 10-8) implying co-precipitation to be effective for strain reduction, as has also been discussed for C and 0 in silicon (see also Chap. 5 of this Volume). Owing to the heterogeneous nature of its precipitation, iron has been used to measure the efficiency of gettering processes involving precipitation, as will be discussed in more detail in Sec. 10.5

10.4.3.2 Copper in Silicon As described in detail in Sec. 10.4.2.2, copper precipitation in silicon is governed by the accommodation of misfit strains which proceeds by self-interstitial emission and their condensation into dislocation loops. Hence any sinks for self-interstitials should catalyze copper precipitation (Seibt, 1992), in agreement with numerous investigations in which copper silicide colonies have been observed on extrinsic Frank-type stacking faults (Seibt 1991; Shen et al., 1994), perfect dislocations punched out from SiO, precipitates (Tice and Tan, 1976; Shen et al., 1994), misfit dislocations (Kissinger et al., 1994), glide dislocations (Gottschalk, 1993; Shen et al., 1996), and grain boundaries (Broniatowski and Haut, 1990; Rizk et al., 1994). Detailed TEM investigations have confirmed that the absorbtion of silicon self-interstitials (emission of vacancies) via dislocation climb is the underlying mechanism of heterogeneous copper precipitation on dislocations. Gottschalk (1 993) observed that dislocation climb due to copper precipitation only occurs on edge segments of glide dislocations or on segments turned into an edge orientation during climb. Later work by Shen et al. ( 1996) confirmed that edge dislocations or edge-type segments of screw dislocations serve as heterogeneous precipitation sites for copper. Both authors argue that the dissociation of 60" dislocations into two Shockley partials constitutes a barrier for dislocation climb, making this dislocation type less favorable for copper precipitation in agreement with their experimental data.

10.4.3.3 Nickel in Silicon Figure 10-24. Rod-like a-FeSi2 precipitate in silicon after in-diffusion of iron at 1 100°C and cooling (i= 1 K/s): the arrow indicates a small precipitate due to residual copper contamination.

Heterogeneous precipitation of nickel in silicon is also frequently observed in the presence of stacking faults (Eweet al., 1994;


Seibt et al., 1998a), dislocations (Lee et al., 1988; Seibt, 1992), and grain boundaries (Rizk et al., 1995). Unlike for copper, considerations of misfit accommodation do not apply for nickel in silicon, as has been shown in Sec. 10.4.2.1. In fact, the possibility of fast incorporation of nickel atoms into the silicide precipitates via the b = a/4( 111) dislocation bounding the platelets was the key to understanding rapid precipitation and the formation of thin Nisi, platelets. Following this guideline, it has been proposed that heterogeneous precipitation of nickel is favored at defects that catalyze the nucleation of the bounding dislocation (Seibt 1992). Considering perfect or partial dislocations, the dissociation reactions

a a a - [i io] + - [i 111+ - [i 111 2 4 4 and

a3 [l 111 + a4 [l 1I] + a [l 111 12

(10-21)

(10-22)

may assist the nucleation process, respectively. The first reaction (Eq. 10-21) has not yet been confirmed experimentally, whereas Nisi, platelet formation at bulk stacking faults is in agreement with Reaction (2) (Ewe et al., 1994; Seibt et al., 1998b). Figure 10-25 is a TEM micrograph showing an Nisi, platelet attached to a bulk stacking fault. In accordance with Eq. (10-22) the two defects form on different ( 1 11] planes since the Burgers vectors of dislocations bounding Nisi, platelets are inclined with respect to the platelet normal (Seibt and Schroter, 1989). It is interesting to note that Ewe (Ewe et al., 1994) observed Nisi2 precipitates exclusively on bulk stacking faults despite a much higher density of SiOz precipitates in his samples (4 x 10" cm-3 compared to 3 x lo', ~ m - ~ ) .

635

Figure 10-25.TypeB NiSi,precipitate atabulkstacking fault in Cz silicon observed after annealing at 1050°C and quenching (f-2000 Us); note that the silicide platelet is inclined with respect to the stacking fault plane in agreement with microscopic considerations (see text) (Ewe, 1996; Seibt et al., 1998b).

10.4.4 Electrical Properties of Silicide Precipitates The electrical properties of silicide precipitates have recently gained interest. This is mainly due to the fact that crystal defects like dislocations (Kittler and Seifert, 1993a; Fell et al., 1993; Shen et al., 19961, stacking faults (Shen et al., 1997; Correia et al., 1995) or grain boundaries (Broniatowski, 1989, Broniatowski and Haut, 1990; Maurice and Colliex, 1989; Rizk et al., 1994, 1995) show extremely high recombination activities when decorated by silicide precipitates. This effect is detrimental especially for multicrystalline silicon materials for solar cell applications which usually contain various defects after crystal growth. Furthermore, silicide precipitates have been shown to be among the few defects in semiconductors that introduce deep bandlike states into the forbidden energy gap. While it is clearly beyond the scope of this article to discuss electronic states resulting from silicide precipitates at crystal defects, this section will briefly summarize the current knowledge of homogeneously formed precipitates. Experimental results

636


on the recombination activity of silicide precipitates as derived mainly from electron beam induced current (EBIC) investigations will be described in Sec. 10.4.4.1. Insight into the nature of deep states at silicide particles has been provided by the analysis of deep level transient spectroscopy (DLTS) data, as is discussed in Sec. 10.4.4.2. It should be noted, however, that these two techniques are usually applied to silicide precipitates at different stages of growth or ripening. Due to the limited spatial resolution of EBIC, only late stages of precipitate ripening are accessible, whereas a high density of small precipitates is needed for DLTS measurements. 10.4.4.1 Recombination Behavior of Silicide Precipitates The electrical activity of crystal defects has often been ascribed to silicide particle decoration. Bearing in mind that secondary defects like dislocations or stacking faults are likely to form during precipitation, the separation of electrical effects due to the two types of defects is rather difficult. In a first study. experimental conditions suitable for nickel precipitation without secondary defect formation were applied to study the recombination properties of Nisi, platelets obtained after quenching and additional annealing at 500-800°C in n-type FZ-Si (Kittler et al., 1991) by means of EBIC. The typical diameter and thickness of these platelets were 1 pm and 5-20 nm, respectively. It turned out that the diffusion length L , of minority carriers as measured by EBIC was governed by the precipitates and depended on the precipitate density N , according to L,=0.7 Furthermore, extremely large EBIC contrasts of up to 40% were obtained. Later investigations of copper silicide colonies have shown EBIC contrasts of up to 93% (Correia et al., 1995). indicating

that silicide precipitates are in general efficient recombination centers. A more quantitative analysis of EBIC contrasts from Nisiz precipitates indicates that potential barriers control the recombination activity of the particles (Kittler and Seifert, 1993b).

10.4.4.2 Spectroscopy of Deep States at Silicide Precipitates Deep level transient spectroscopy (DLTS) is a well-established technique to study deep levels associated with isolated point defects in semiconductors. For extended defects, broadened DLTS lines (Kimerling and Patel, 1979; Omling et al., 1985) as well as nonexponential capture kinetics (Figielski, 1978; Wosinski, 1990; Gelsdorf and Schroter, 1984; Grillot et al., 1995) havefrequently been observed. They can be consistently interpreted if in addition to the occupationdependent capture barrier internal transitions between the states (although not contributing to the measured signal) are taken into account (Schroter et al., 1995). The internal equilibration time can be used to classify extended defects into bandlike ( K - e R ; ' , R,' ) and localized (T'PR;', R;') states, where Re and R, are the emission and capture rates of the defect, respectively (Fig. 10-26). As a result, on the timescale of DLTS experiments, the total occupation F of bandlike states is given by a Fermi distribution with a time dependent quasi-Fermi energy, whereas localized states exclusively exchange charge carriers with the conduction and the valence band. Experimentally, the two limiting cases can be distinguished by the different dependence of their DLTS lines on the filling pulse duration, tp (Schroter et al., 1995): For bandlike states, the high temperature sides of DLTS lines coincide for different t p values.

637

10.4 Precipitation of Transition Elements in Silicon 71

'

'

"

" " " " ' J

"

6A

r

0

9 0 a

.

5-

4:

321 n ",

Figure 10-26. Band diagram for electronic states at extended defects; 6E,= a [ F - F c n ) ] denotes the capture barrier of defects for which the occupation probability F is larger than for the neutral defect (Fen'); 4 is the internal equilibration time which can be used to classify deepstatesinto bandlike(44Ri1,R;') and localized (T,*R;', R;'), where Re and R, denote emission and capture rates, respectively.

For localized states, the high temperature sides of DLTS lines coincide if spectra are normalized with respect to their maxima. Nisi, platelets obtained after quenching from high temperatures give rise to DLTS spectra as shown in Fig. 10-27a. From the common high temperature sides, it can be concluded that bandlike states are associated with the platelets. From the atomic structure of the platelets (Sec. 10.4.2.1), two possible origins of these states can be inferred, i.e., (1) the Nisi,/( 11 1)Si interfaces, and (2) the bounding dislocation. Numerical simulations show that a distinction between these possibilities on the basis of DLTS data alone is not possible although the densityof-states @ ( E )= const for two-dimensional systems and @ ( E )cc 1/V%for one-dimensional systems are considerably different. Making use of the internal ripening phenomenon associated with Nisi, platelets, Riedel and Schroter (2000) could show that bandlike states transform into localized states during this process (Fig. 10-27b). The fact that this transformation occurs on a much shorter time scale than structural changes revealed by HRTEM has been tak-

140

100

a)

A

0

% 0

.

9

r!

3

220

260

300

1

T (K) 1.1 I 1.0:

'

'

,

1

0.9 : 0.6 ; 0.7 ; 0.6 : 0.5 0.4 ; 0.3 ; 0.2 ; 0.1 : 0.0 c

50

b)

180

170 190 210 230

250

270

290

T (K)

Figure 10-27. DLT spectra of Nisi, platelets obtained with different filling pulse durations tp: a) platelets obtained after quenching from 900°C (T-2000 Ws)introduce bandlike states into the band gap of silicon, as is deduced from the common high temperature side of DLTS lines, b) internal ripening of precipitates results in a transformation into localized states, as revealed by the common high temperature sides of normalized spectra.

en as evidence for the bounding dislocation to be the origin of the deep states associated with Nisi, platelets (Riedel and Schroter, 2000). These authors argue that structural transformations during internal ripening are initiated by climb of the bounding dislocation out of the platelet plane, destroying the translational symmetry within its core and thus the bandlike nature of the states. For copper, deep bandlike states have been observed after quenching from high temperatures (Seibt et al., 1998a; Istratov et al., 1998b; Sattler et al., 1998; Flink et al., 1999a). The associated DLTS lines are con-

638


siderably broader than those for the Nisi, platelets shown in Fig. 10-27 and are consistent with the density-of-states function of two-dimensional systems [e( E )= consf]. Although a fit of DLTS lines to numerical solutions of the rate equation has not yet been possible, a rough estimate yields an energy band with the center at E,-0.25 eV and a width of 0.20eV. An important result seems to be the large neutral occupation of 0.8 I F'"'I 1.O, which indicates that copper silicide precipitates obtained after quenching are positively charged unless the Fermi level is above E,- (0.15 - 0.19) eV. Bearing in mind that interstitially dissolved copper is positively charged (Sec. 10.2.2), Coulomb repulsion between Cuj+' and the precipitates is expected. In fact, a charge-dependent precipitation behavior has recently been observed (Flink et al.. 1999a, b) for interstitial copper which diffuses out of silicon wafers or precipitates in the bulk if the Fermi level is below or above E,-0.20 eV, respectively, which roughly coincides with the neutrality level of copper silicide precipitates. The origin of the deep bandlike states has not been unambiguously identified by DLTS. Structural changes observed during internal and Ostwald ripening result in changes of the observed DLTS line shape and amplitude, while the bandlike nature of the deep states is conserved. This has been taken as evidence that the states are not due to secondary defects involved in copper precipitation, since their structure changes during ripening (Seibt et al., 1998a; Seibt et al., 1999). Hence the deep states may tentatively be attributed to precipitatelmatrix interfaces or the precipitates themselves.

10.4.5 Silicide Precipitation at Si/Si02 Interfaces Owing to the fact that the dielectric properties of SiO, films are of utmost importance

in silicon device technology, the tendency of fast diffusing transition elements to form precipitates at wafer surfaces (i.e., Si/SiO, interfaces) gives rise to serious problems. For example, iron silicide precipitates are known to degrade reverse junction breakdown characteristics of bipolar transistor devices (Cullis and Katz, 1974; Augustus et al., 1980). In the following we shall focus on the influence of silicide precipitates on the performance of gate oxides. It has been shown that one of the main problems concerning the dielectric quality of SiO, films is due to local decomposition into Si and volatile S i 0 (Rubloff et al., 1987). This decomposition is catalyzed by metallic impurities or decorated stacking faults at the Si/SiO, interface (Liehr et al., 1988 a, b). In a series of TEM studies, Honda and co-workers demonstrated that nickel, copper, and especially iron silicide precipitates located at the Si/Si02 interface lead to breakdown voltages with an increasing effect from nickel to iron (Honda et al., 1984, 1985, 1987). Recent TEM investigations provided further evidence for the formation of b- FeSi, precipitates during dry oxidation (Wong-Leung, 1998). In these experiments, iron impurities were introduced as a surface contamination prior to oxidation. The authors further observed a correlation between precipitates and sharp asperities at the interface apparently consisting of silicon. Although their origin has not yet been clarified they may be a result of local reduction of S O , catalyzed by FeSi, precipitates. For copper, cross-sectional TEM revealed that silicide precipitates located at the Si/SiO, interface protrude into and hence thin the oxide (Wendt et al., 1989; Cerva and Wendt, 1989b) which leads to a poor quality of gate oxides in MOS devices. Conversely, Nisi, platelets at Si/SiO, interfaces do not grow into the oxide; this is

10.5 Gettering Techniques and Mechanisms

correlated with high breakdown voltages (Cerva and Wendt, 1989a). Thus it has been demonstrated that the growth behavior of silicide precipitates and electrical effects in device technology may be closely related.

10.5 Gettering Techniques and Mechanisms 10.5.1 Introduction to Gettering Mechanisms Gettering means the removal of impurities from the device-active area by transport to predesigned regions of the wafer or by evaporation. The latter is termed chemical gettering, which makes use of atoms (e.g., Cl) attacking the silicon surface to form volatile complexes with transition elements [for a review, see Shimura (1988)l. The definition of gettering excludes passivation as a technique or mechanism because it is not the impurity that diffuses, but an additional impurity to form an electrically inactive complex [see, e.g., reviews on hydrogen in silicon: Pearton et al. (1987, 1989)]. With respect to gettering techniques in silicon, there is a distinction between internal and external gettering (also referred to as intrinsic and extrinsic). External gettering means the gettering of impurities by extrinsic (P, B, oxygen precipitates, etc.) or intrinsic defects (Siself interstitials, stacking faults, dislocations) after an external treatment, e.g., P-diffusion (Meek and Seidel, 1975; Lescronier et al., 1981; Cerofolini and Ferla, 1981; Shaikh et al., 1985; Falster, 1985; Kuhnapfel and Schroter, 1990), ion implantation (Meek and Seidel, 1975; Myers et al., 1996), mechanical abrasion (Mets, 1965), poly-Si deposition (Chen and Silvestrii, 1982; Falster, 1989; Girisch, 1993; Seibt et al., 1998b; Hayamizu et al., 1998), AI-Si liquid gettering (Seibt et a]., 1998b; Tan et al., 1998). In

639

the case of internal gettering, extrinsic or intrinsic defects are already present in asgrown material or they are produced and activated by an annealing treatment, like oxygen precipitation gettering (Tan et al., 1977) or p/p+ gettering (Hayamizu et a]., 1998). Most of the external gettering techniques require a treatment of the back side, whereas internal gettering utilizes the bulk of the wafer close to the device-active region. In the literature, a class of gettering techniques is termed proximity gettering. These are techniques that have gettering sites in the immediate vicinity of the devices (closer than the denuded zone of oxygen precipitation gettering), or are even part of the device itself (like Si/Ge, frontside ion implantation). Basically, two types of gettering mechanism are conceivable. One type requires supersaturation of metal impurities as the driving force for precipitation and preferred nucleation in the gettering layer. A high density of heterogeneous nucleation sites in the gettering layer enables metal impurity precipitation faster and at smaller supersaturations in the gettering layer than in other parts of the wafer, resulting in a concentration gradient of the diffusing metal impurity species towards the gettering layer. Outside the gettering layer, the probability for nucleation is reduced. This mechanism is called relaxation gettering, because the relaxation of a supersaturated solution by the nucleation and growth of precipitates at defect sites is used to lower the metallic impurity content of the device-active part of the wafer. Other gettering mechanisms do not require supersaturation. Such gettering action may come about by an enhanced solubility of metal impurites in the gettering layer or by stabilization of a new metallic compound in the gettering layer. This mechanism is termed segregation gettering. Some effort has been put into finding experimental evidence for the mechanism operating in a

640


given technique. A convincing experimental design to distinguish between segregation and relaxation gettering is the single step annealing of a wafer with an impurity concentration below the solubility and termination of the annealing step by quenching such that RMeLo, where Lo is the distance between the device-active area and the gettering layer and RM is the diffusional range of the impurity M (see below). If the total impurity concentration is increased inside and decreased outside of the gettering layer, evidence is presented that the specific gettering technique investigated did not require supersaturation of the metal impurities. This direct proof of segregation gettering was established for PDG, A1 gettering, and gettering by cavities (“cavity gettering”), as will be discussed i n Sec. 10.5.3. In order to verify that a given technique is operating by relaxation, two-step annealing and quenching to room temperature is necessary. During the first step, the contamination at or below the solubility is distributed homogeneously within the wafer. With the second step, at a temperature low enough to create supersaturation and by quenching to room temperature RMQ Lo, it can be verified from a measurement of the total impurity concentration outside and in the gettering layer that just the impurity amount in supersaturation has been gettered, Le., the concentration of the impurity species has dropped to the solubility limit at the low temperature step. The diffusional range RM introduced above may simply be estimated as RM= d 6 D, ( TG)tG for isothermal gettering annealing at the temperature T , for an annealing time t G . Gettering techniques based on the relaxation mechanisms mainly operate during cooling from high temperatures. In this case, the temperature T(sup)at which the metallic impurity becomes supersaturated with respect to its precipitate and the temperature-time profile, T, (t), during cooling

become important experimental parameters. The diffusional range may then be estimated as

\dtl

J

(1 0-23)

Taking Tkup)=900°C (800”C)and(dT,ldt) = 1000 Ws, gives the following R , values: Fe: 11.5 (7.8) pm, Co: 54 (42) pm, Ni: 47 (34) pm, Cu: 42 (36) pm. For a different value x of the cooling rate, it is necessary to multiply the given values of RM by (lOOO/x)”’. A necessary condition for relaxation gettering to operate is RM>Lo. In silicon wafers, which contain several types of extended defects with different mean distances, RM may be chosen for selective precipitation at the defect with the highest density or, if another extended defect presents a stronger nucleation site, for precipitation at this one (see Sec. 10.4.3). If RM is much larger than the mean distance between the possible nucleation sites, ripening is expected to occur and to yield fewer and larger precipitates. In the following sections, we will describe the mechanism of internal gettering (Sec. 10.5.2) and then outline the present understanding of the major external gettering techniques, poly-Si backside gettering (Sec. 10.5.3.l ) , aluminum gettering, and cavity gettering (Sec. 10.5.3), and phosphorus diffusion gettering (PDG) (Sec. 10.5.3.2). 10.5.2 Internal Gettering 10.5.2.1 Oxygen Precipitation Gettering Oxygen Precipitation and Denuded Zone Formation The concept of internal gettering was first introduced by Tan et al. (1977) to improve

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the so-called leakage-limited yield of bipolar transistors in Cz-Si. The authors argued that metal impurities precipitate at dislocations punched out by silicon oxide precipitates (Tice and Tan, 1976). Due to the technological importance of Cz-silicon, the use of oxygen precipitation-induced defects as efficient gettering centers has stimulated a great deal of investigation, and many details about oxygen in silicon have been uncovered [for reviews, see Corbett et al. (1989) and Borghesi et al. (1995)l. Oxygen is incorporated into single crystalline silicon grown by the Czochralski method by reaction of the Si melt with the quartz crucible. Typical oxygen concentrations are 5 x 1017 to 1x 10l8 atoms/cm3 and can be adjusted with an accuracy o f f 10%. For processing temperatures of lOOO"C, the oxygen solubility is about 1017 atoms/cm3 (Hrostowski and Kaiser, 1959; Mikkelsen, 1982a, b). Internal gettering makes use of defects resulting from the formation of silicon oxide precipitates in silicon. This process is associated with a large volume expansion which is relaxed by the emission of silicon self-interstitials (leading to stacking fault growth) or by plastic deformation of the silicon matrix (prismatic punching) [for reviews, see Hu (1981) and Bourret (1986)l. For internal gettering, special annealing cycles have been designed which allow the creation of oxygen precipitation-induced defects in the bulk of silicon wafers, but not in the device-active area. Although modern processes for the fabrication of semiconductor devices (integrated circuits) have greatly simplified the activation of internal gettering by modification of the temperature ramps, the basic principle is still the same, and can be described by a three-step annealing sequence: Wafers are first treated at a temperature (usually above 1050°C) that is sufficiently high to prevent the formation of

641

stable nuclei and allow out-diffusion of oxygen from a subsurface layer with a thickness of several micrometers. In contrast to the bulk of the wafer, the oxygen supersaturation in this area is not sufficient to initiate oxygen precipite nucleation during a subsequent low temperature treatment (usually about 700°C). Finally, a second high temperature annealing (about 1050"C) is used to initiate oxide precipitate growth and the formation of secondary defects. Figure 10-28 shows the resulting defect distribution as obtained by preferential etching of a wafer cross-section: Below a several micrometers thick denuded zone, bulk microdefects are clearly visible. Gettering Mechanism of Interstitial Impurities

Many investigations have aimed to clarify the gettering mechanism for interstitial impurities by oxygen precipitates and related extended defects. The results demonstrate that extended defects (oxide precipitates, stacking faults, dislocations) are decorated by metal precipitates, identifying possible sinks for gettering. Falster et al.

Figure 10-28. Cross section of a wafer that was defect-etched to reveal extended defects after the internal gettering process (magnification: 5 0 ~ )Denuded . zone and bulk microdefects are clearly visible (with kind permission of Wacker Siltronic AG, Burghausen).

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642


(199 1 ) found oxygen precipitates decorated by copper and nickel precipitates. The shape of the metal precipitates depends on the cooling rate and changes during a second annealing step, which is interpreted as the release of metal impurities upon annealing. Shen et al. (1995) chose annealing conditions in Cz-Si that resulted in oxygen precipitation accompanied by the formation of stacking faults and dislocation loops. The authors found that Fe, Ni, and Cu decorate Frank partial dislocations of stacking faults, the amount of which depends on the cooling rate. Ewe et al. ( 1 994) found the decoration of stacking faults, but not of oxygen precipitates, although the precipitate density exceeds that of stacking faults by a factor of 100. Although at a first glance some of the results might appear conflicting, the experimental conditions with respect to pretreatment of Cz-Si, the contamination level, and the cooling rates are usually quite different. In fact, these observations demonstrate the significant role of the diffusion range R, of an impurity as a selection parameter for inhomogeneous precipitation. Shen et al. (1995) have studied the precipitation of Fe, Ni, and Cu in Cz-Si containing oxygen-precipitation induced stacking faults bounded by Frank partials (FPs) of a density between 1.2 x lo7 and 8.5 x 10’ cm-’, corresponding to mean distances of between 1 1 pm and 44 pm, and punched out dislocation loops of a much lower density. After indiffusion of the metallic impurity, they applied two rather different rates of 1600 Ws and 23 Ws to quench the wafer. The decoration of Frank partials (estimated from EBIC contrast) by Fe increases with R,, changing from 17 pm for 1600 Ws to 140 pm for 23 Ws. Ni and Cu form precipitates at the Frank partials for RNi=47pm and Rc,=93 pm, and large Ni-precipitates and even colonies of Cu-precipitates away from the Frank partials, probably nucleated

at the punched out dislocation loops for RN,=392 pmandR,,=784 pm. Withadifferent diffusion temperature of 1200°C and a quenching rate of 100 Ws Falster et al. ( 1 991) found a similar precipitation behavior of Ni and Cu. Summarizing the results, evidence has been presented that extended defects caused by the precipitation of oxygen act as nucleation sites for metallic impurity precipitation. In some cases, the precipitates have been identified as the equilibrium silicides (FeSi,, Nisi,, Cu’Si). With respect to classification of the mechanism, Aoki et al. (1992) have shown for Fe in Si, that 1. for slow cooling, the concentration of interstitial Fe in the near-surface region is reduced in Cz-Si after oxygen precipitation by up to an order of magnitude compared to as-grown Cz and Fz-Si for a high contamination by Fe (2 x lOI4 atoms/ cm’), but not for a low contamination by Fe (3 x 10” atoms/cm3); 2. after quenching, there is no reduction of the initial Fe concentration in the near surface region for all samples, i.e., for low as well as high Fe contamination; 3. for prolonged annealing of those samples at a temperature at which the Fe concentration exceeds the solubility, the residual Fe-concentration is reduced and, depending on the length of the annealing time, approaches the solubility at the annealing temperature.

These experiments clearly demonstrate that all criteria for relaxation gettering are fulfilled. Gilles et a]. (1990b, c) established a quantitative correlation between oxygen precipitates and nucleation sites for iron precipitation. They applied Ham’s theory (Ham, 1958) of diffusion-limited precipitation from a supersaturated solution to the precipitation kinetics of oxygen and iron.

10.5 Gettering Techniques and Mechanisms 10

They determined the product n x ro of both ( n and ro are the density and radius of the precipitates, respectively). The time constant of the exponential precipitation kinetics can be expressed as t=

1 4rcDnro

643

0

(1 0-24)

where D denotes the diffusion coefficient of the diffusing species. By applying Eq. (10-24) to the precipitation kinetics of oxygen, the size and the density of oxygen precipitates can be determined separately. Approximating the precipitates by spherical particles and their radius by the maximum value after precipitation, the loss of interstitial oxygen with annealing time, as measured by infrared absorption spectroscopy, determines the product n x r i . Livingston et al. (1984) reported good agreement for oxygen precipitate densities calculated this way (Binnset al., 1983, 1984) with those obtained after the application of a defect-delineating etch or by smallangle neutron scattering. Figure 10-29 shows a plot of the iron precipitation kinetics at 235 "C in Cz-Si pre-annealed at 700 "C to nucleate oxygen precipitates and then annealed at 1050 "C for 2 h to saturate the specimen with Fe. For longer pre-annealing times, a higher density of oxygen precipitates is expected after the high-temperature step. Fast quenching was chosen to keep Fe on interstitial sites. The concentration of interstitial Fe was determined by the electron paramagnetic resonance technique. By variation of the temperature, it is found that the precipitation kinetics is thermally activated with an activation energy of 0.7 eV, which is in good agreement with the migration barrier of interstitial Fe. Hence the application of Ham's theory modeling diffusion-limited precipitation is justified.

0

-50

100

50

150

Annealing time in min

Figure 10-29. Precipitation kinetics of Fe in silicon at 235 "C after thermal processing (700°C, 4 h (47 h, 64 h, 192 h) and 1O5O0C,2 h) and quenching. FZ denotes floating zone silicon with an oxygen concentration below the detection limit of FTIR. The other data were obtained from Cz-Si ([PI = 1.5 x 10l6 atoms/ cm3, [Oil = 1.I x 10'' atoms/cm3 (IOC88-standard) and [C] = 1.5 x 10l6 atoms/cm3). Preannealing time at 700°C is indicated in the figure.

An analysis of the precipitation time constants of oxygen and Fe is shown in Table 10-9, in which known values of diffusivities and measured values of the time constant were inserted using Eq. (10-24) to determine n x r: for oxygen as well as for iron.

Table 10-9. Comparison of the products of density n and size ro for oxygen precipitates (calculated from the loss of interstitial oxygen, Eq. 10-24) and nucleation sites for Fe precipitation (calculated from the time constant of Fe precipitation at 235 "C) after thermal processing (7OO0C, 4 h (32 h, 128 h) and 950°C, 4 h (lO5O0C, 2 h)). Preanneal time (h)

4 32 128

Diffusion temperature 950°C 1050°C n ro ( 0 ) n ro (Fe) ( io4 cm-2)

n ro (0) n ro (Fe) ( io4 cm-*)

8.6 25 75

1 is built up (Fahey et al., 1984) as shown schematically in Fig. 5-19 (see also Fig. 10-31). The main source of I is the dissociation of P-I pairs near to the transition between the kink and tail region (Morehead and Lever, 1986; Mulvaney and Richardson, 1987; Orlowski, 1988). The released silicon interstitials partly diffuse into the wafer, generating a stationary supersaturation and giving rise to enhanced diffusion of P in the tail region, and partly diffuse back to the PSG/Si interface, which is considered as an I sink.


650

\

,.."........................................

[I1 [I],

1-

-

e

I

I1

M,W+ e- 4+ M,((.+lf-)

[VI,

4

'

iht'"s'

depth

Figure 10-31. Schematic presentation of concentration profiles and associated fluxes (arrows) of P, I, and V during P-indiffusion. The coupling by reaction between P and I gives rise toj,, that between I and M toj$'J). The coupling which i n d u c e s j F g r lis the pairing between M, and P and the electron transfer from P to the acceptor states of M,, whereby [M,],, increases (Fermi level effect). Vacancies that may react with self-interstitials by I + V 0, where 0 stands for an undisturbed lattice site (see Sec. 5.3 of Chap. 5 of this Volume), are expected to be in undersaturation in the bulk and attain their equilibrium value [VI,, at the surface. Please note that if the I or V introduces an acceptor level into the band gap, which for high temperatures is not known, its equilibrium concentration would increase in the highly P-doped region.

Gettering of a metallic impurity M under the conditions of P-in-diffusion has a segregation component that manifests itself through the dependence of the equilibrium concentration [MI,, on the electron concentration (Fermi level effect) and on the phosphorus concentration (M-P pairing) (Hall and Racette, 1964; Cagnina. 1969; O'Shaughnessy et al., 1974; Meek and Seidel, 1975; Gilles , a second et al.. 1990). For Pcsurf)> [PIccrit) component, called injection-induced, connects the flux of M to the supersaturation of silicon interstitials, [I]/[I],!, and to the gradient [I]/[I],, through the kick-out reaction: I + M, F! Mi (Schroter and Kuhnapfel, 1989, 1990). If local equilibrium between M,, M,, and I is established everywhere in the specimen and at any time through this reaction (10-26)

It has been shown that the concentrations in Eq. (10-26) represent total ones, i.e., sums over all charged species of Mi, M,, and 1. Also pairs like, e.g., M,P may be included in [M,] if local equilibria of electron exchange and pairing may be assumed (Spiecker et al., 1997). Since the time scale of PDG is the rather slow in-diffusion of P, electronic, pairing and kick-out reactions may be taken as being in local equilibrium. Equation ( 10-26) already implies that increasing the concentration of self-interstitials increases the ratio rM=[Mi]/[M,]. As can be seen from Fig. 10-31, the supersaturation of silicon interstitials [I]/[I],, under P-in-diffusion is largest outside the highly P-doped region of the wafer, decreases with increasing P-concentration, and approaches unity at the wafer surface. According to Eq. (10-26) rM/rM,eq varies in the same manner. As a consequence, a gradient of the mobile interstitial component of M will build up in


the highly P-doped region and drive a current of M towards the wafer surface. This gettering contribution has been called injection gettering, and we note that it is associated with any gradient of intrinsic point defects that react or interact with mobile impurity species. Under P-in-diffusion, the injection-induced current of M is expected to establish a major contribution to gettering if rM,eq=[Mi]eq/[M,]eqdl in the intrinsic and in the P-doped region of the wafer. This is the case for Au, Pt, and Zn. For the 3d elements, rM,eqS1 in intrinsic silicon and rM,eqQ1 in highly P-doped silicon (Gilles et a1.,1990a), so that [I]/[I],, can only slightly modify [Mi] in the bulk, but might have a significant effect on it in the highly Pdoped region of the wafer.

Segregation and Injection Gettering (Diffusion Mode) In this section, we summarize the present understanding of the PDG mechanisms operating for [P](suf15[PI,, and for Au in Si. The Si-Au system has been carefully investigated by Sveinbjornsson et al. (1993). They have obtained a set of measured [PI profiles and associated [Au] profiles. These results have proved to be an excellent basis for developing and critically checking a gettering model comprising segregation and the effect of silicon interstitial injection by P-diffusion (Spiecker et al., 1997). The modifications needed when applying this model to Pt and 3d elements are discussed at the end of this chapter. The only species of M that contributes to transport in intrinsic silicon is the interstitial one, Mi. As mentioned above, Mi of most metallic impurities is in its neutral charge state in intrinsic and in n-type silicon, so that [Mi]= [Mi”] (see Sec. 10.3; an exception is Cui, which is positively charged). The flux of M, j,, is then deter-

651

rul

M

segregation + injection

segregation

(b)

depth

Figure 10-32. Schematic presentation of the segregation coefficient S and the supersaturation of silicon interstitials [I]/[I],,, and a) their effect on the distribution of gold and b) inside the silicon wafer during PDG.

mined by D i , the diffusion coefficient of Mp (10-27)

j M can be decomposed into three contributions, i.e., one diffusion current balancing the concentration gradients of M, and two drift currents, which result from segregation and I injection associated with P-diffusion (10-28) The analytical form of these contributions is obtained easily by inserting Eq. (10-26) into Eq. (10-27) and taking into account, that the total concentration o f M is given by [MI = [Mi]+ [M,], where [M,] is the sum over all charged species and pairs of M,. The first component of Eq. (1 0-28) is for a diffusion current, which follows the neg-

652


sion coefficient is Di rM,,, for [I]/[I],,= 1 ative gradient of [MI and tends to equalize and changes to Di for rM,,,. [I]/[I],,Sl. its concentration throughout the wafer. The Silicide precipitates like Cu,Si, which emit two other components are drift currents, silicon interstitials when growing, become which are proportional to the total concenunstable for [I]/[I],,> 1 (bulk of the wafer) tration of M ([MI) and follow the gradient compared to regions where [I]/[I],,= 1 (waof the segregation coefficient Sand the negfer surface region). The opposite holds for ative gradient of the silicon interstitial precipitates like FeSi,, which absorb intersupersaturation, [I]/[I],, , respectively. The stitials or emit vacancies when growing, two drift components operate in an opposite while the stability of Nisi, and COS, predirection to the diffusion current, as is cipitates is expected to be almost indepenshown schematically in Fig. 10-3 1. The local segregation coefficient compares dent of [I]/[I],, (see Table 10-8). the total solubility of M at a given position For the quasi-steady-state, an approxiwith that in intrinsic material, S:=[M],,/ mate solution of Eq. (10-28) for [MI ( x , t ) may be obtained with j M = O (Spiecker et al., [M]eq,intr. Please note, that the solubilities [...Ieq and have to be taken for 1997). The M-current components due to [I]/[I],, = 1, so that S is independent of the diffusion, segregation, and injection, whose silicon interstitial supersaturation. The posum yields jM(see Eq. (10-28)) are very larsition dependence of S results from the varige compared to j, in the highly doped reation of [MI,, with the phosphorus concengion. In the same way that the balance betration, so that i t is finally the P-concentratween diffusion and drift current in a pn tion gradient that drives the segregation junction, i.e., the condition jpn=O, is applied component of the M current. to calculate the internal electric field, j M = O Segregation gettering is the only compois used to calculate the internal M concennent observed for [P]csurf'c [PIccrit).For tration field. This solution then allows cal[P](su6'>[PIccritJ, in addition to the culation of the gettering efficiency G, which silicon interstitial current towards the suris the ratio of the concentrations of M at the face induces the M-current component j$JJ surface and in the volume in the same direction. This injection-induced current is driven by P-in-diffusion, ( 10-29) wherefore we propose to name the associated gettering mechanism the diffusion mode of injection gettering. In the next section, we will present evidence that for Pcsu6,>P,, a further injection mode associated with Sip particle growth may also lead to gettering. The I supersaturation also affects the kinetics of gettering (Spiecker et al., 1997) Segregation alone yields G = S(surf).For and the stability of silicide precipitates weak I injection, so that ( [I]/[I],,)'b"'k)4 (Bronner and Plummer, 1987). The transG= ([I]/[I],, )(buk) x S(sum ([MI,,/ [Mileq port coefficients of the three M-current is obtained, and for strong I injection G = components are dependent on [I]/[I],, and ([M]eq/[Mi]eq)(intr)~S'surf). rM,,,. This influence is most pronounced for Let us take gold in silicon as an example, impurities like Au, Pt, and Zn, for which for which ([M],,/[Mi]eq)(inh.)=130 at 900°C rM,,,+ 1, and for which the effective diffuhas been estimated by Boit et al. (1990). The

jggr),


value of ([I]/[I]ep)(bU1k) is dependent on the conditions of P-in-diffusion. Taking a value of 102,G = 57 S(surf) is obtained, and for a Experimental value of lo3, G=l15 S(surf). values of G, measured by Sveinbjornsson et al. (1 993) for Au in Si at 900 "C, are in the range of our rough estimation. A more detailed and quantitative comparison between experimental data and predictions of the gettering model requires the input of S(x, t ) and [I]/[I],, (x,t ) from a Pdiffusion model. Such a comparison has been performed recently for the WAu system and has demonstrated very good agreement. Au-profiles measured by Sveinbjornsson et al. (1993), especially in the highly P-doped region and the associated P-profiles have been almost exactly reproduced by the simulations (Kveder et al., 2000) (see Fig. 10-33). The model is also expected to be applicable to other metallic impurities with apredominant substitutional component and a

highly mobile interstitial one, like Pt and Zn (Schroter and Seibt, 1999a). Pt is the left neighbor of Au in the 5d row of the periodic table. The substitutional species of both elements have a donor and an acceptor level in the band gap of silicon and show similar diffusion profiles evidencing the effect of the kick-out reaction. Differences between Au and Pt in silicon appear when the phase diagrams are compared: Si/Au is a simple eutectic, while SiPt forms several silicides. While silicide formation has been found after PDG with [PI,,, probably caused by a new mode of injection gettering, its role for Pt in PDG with [P](~"'OI [PI,, is not clear at present. The action of the kick-out mechanism on M, drives injection gettering. Its direct action on 3d elements with a predominant interstitial component in intrinsic silicon is expected to be rather small. However, PDG experiments at 920°C for 54 min { [ C O ] ( ~ ~ ~ ~ ~ ~ ~ 4 x 1014~ m - [cole,= ~, 1.I x 1014crnp3,spec-

1 om

1 oZ0

1 ol*

10'

p 1ol8

10"

E 1017

10'~

8e

10'8

1 ole

10''

lo=

io"

10l4

1 013

10'~

1 o'*

1 0l2

-1 E

0"

653

10"

10" 0

1

Depth [ w l

2

0.0

0.5

1.0

1.5

2.0

Depth [rml

Figure 10-33. Simulation (Kveder et al., 2000) and experimental data (Sveinbjornsson et al., 1993) of PDG for Au in Si. The initial gold concentration was uniform at 3 ~ 1 0cm-3 ' ~ and below the solubility of Au at 980"C, which is 8 . 6 1015 ~ atoms/cm3. The points show the experimental SIMS profiles for P and Au after PDG. A) PDG at 980°C for 30 min followed by slow cooling down to 900°C with a rate of 5 Urnin, then fast cooling; B) additional annealing of the same sample at 1150"C, for 15 min without P exposition and quenching. The phosphorus glass was removed from the surface before the second annealing.

654


imen quenched with arate of about 1000 Us after PDG) have clearly shown, that concentrations of Co in the highly P-doped layer are of the same order as those found = ~ cm-3, [ C O ] ( ~ " ' ~ ) = for Au { [ C O ] ( ~ " ~x '10I8 10l3~ m - (Kiihnapfel, ~ } 1987). This puzzle has not really been solved yet, but independent experimental findings have indicated possible solutions. Studying solubility and diffusion of Mn, Fe, and Co in P-doped silicon and including literature data on Cu in their analysis, Gilles et al. (1990a) found that the ratio r = [Miles/ [M,],, for these elements drastically decreases from the value in intrinsic silicon (rS-1) to r=l and further to r 6 1 , when the P-concentration becomes larger than the intrinsic electron concentration (see Sec. 10.3). These results establish the prerequisite for the injection-gettering mode to operate in the limited region of high P-doping, but they also imply the possibility of a stronger segregation-gettering by the multiple acceptor action of Cos. Since the model developed for Au in Si and described above comprises both mechanisms, numerical simulations and comparison with available experimental results for Co in Si are expected to extend the validity range of the model to 3d elements. Figure 10-34. (a) Lattice image of PSG/Si interface Segregation and Injection Gettering (Precipitation Mode) For [P]csurf)> [PI,, experimental results, especially from TEM, indicate a new mode of injection gettering which is driven by local gradients of [I]/[I],, and is associated with the incorporation of inactive P into the silicon wafer occurring simultaneously with P-in-diffusion. We propose to call it the precipitation mode of injection gettering. It has been detected by the fact that silicide precipitates grow from the PSG/Si interface near to Sip needles. In this section, we brief-

in the neighborhood of a Sip particle growing into the silicon. Note the protrusion of the Sip particle and of the PSGlSi interfacenearthe particle.Theundisturbed interface is indicated by the dotted line. (b) Brightfield TEM image of the PSGlSi interface showing a Sip particle and a precipitate identified as Nisi, (according to Ourmazd and Schroter, 1985).

ly outline the main feature of the precipitation mode and silicide formation. For [PI,,, Sip precipitates have been observed to grow from the Si/PSG interface (PSG: phosphorus silicate glass) into the silicon bulk (Bourret and Schroter, 1984). For every Si-atom, that becomes in-

10.6 Summary and Outlook

corporated into the Sip particle, 1.5 silicon interstitials have to be injected into the silicon to adjust the difference of the silicon specific volumes. As a result, a current of silicon interstitials is generated at the progressing SiP/Si interface. Since epitaxial growth of silicon has been observed at PSG/Si interfaces near to the Sip precipitates (see Fig. 10-34a), it has been argued that some part of this I current is directed towards this region of the interface. In the presence of substitutional metallic impurities, the local silicon interstitial current is expected to induce a local impurity current towards the PSG/Si interface. This local configuration might be considered as a small pump of injection gettering. It is local in the sense that its operation is limited to the region of high P-doping. Consequently, it should act on all impurities where r s l , Le., also on 3d elements. Indeed, after PDG of Nidoped wafers, Nisi, precipitates have been observed at the SiP/PSG and the surrounding SiPSG interface by TEM (Ourmazd and Schroter, 1984) as is shown in Fig. 10-34b. Recently, Pt has also been found after PDG with Sip growth as the orthorhombic PtSi precipitate at Sip (Correiaet al., 1996). Modeling and numerical simulation of the local precipitation mode of injection gettering associated with Sip growth and silicide formation are open problems at present.

10.6 Summary and Outlook In this chapter we have outlined the high temperature characteristics of those transition elements in silicon, that have been studied in some detail. These are the 3d elements from Ti to Cu, the 4d element Pd, and the 5d elements Au and Pt. The solubility and diffusivity of 3d elements and Au have been investigated in intrinsic and extrinsic silicon. In intrinsic silicon, all 3d elements

655

and Pd dissolve predominantly on tetrahedral interstitial sites, which means that [Mileq> [M,],, , while Au and Pt mainly dissolve substitutionally, so that for them , Cui, [Mileq < [M,],, . For Mni, Fei, C O ~and the charge state of the dominant species has been determined to be MIo’. For temperatures below 1100 K, the interstitial species of Mn, Fe, Co, and Cu have been shown to be donors, the substitutional ones to be multiple acceptors. Consequently, in extrinsic silicon [M,],, > [Mileq in this temperature range. For Mn,, Fei, and Coil a strong shift of the donor level towards the valence band above 1100 K indicates a transition from a low temperature to a high temperature atomic configuration. Compared to the usual solubilities in metallic systems, partial solution enthalpies found for the transition elements in silicon are very large (1.5-2.1 eV). Coi, Ni,, Pd,, and Cui are among the fastest diffusing impurities in silicon with migration enthalpies below 0.5 eV. For the lighter 3d elements, the diffusivities decrease and AH?’ increases to about 1.8 eV for Ti,. The systematics and for the heavier 3d elements also the absolute values of these interstitial diffusivities have been explained by a simple hard sphere model. If the concept of atomic radius is transferred to these metallic impurities in silicon, the difference of the elastic energy between the tetrahedral and the hexagonal site has been found to be a major contribution to the migration enthalpy of diffusion. Concerning the precipitation behavior of the fast diffusing transition elements cobalt, nickel, copper, and palladium, there is now some detailed knowledge as to which precipitate structure and composition forms under various experimental conditions. We have seen that precipitation of these impurities is closely related to the more general question of how systems with large driving forces relax toward thermal equilibrium.

656


Apparently, kinetically selected structures are initially formed. They transform into energetically more favorable configurations during Ostwald ripening or internal ripening, a process closely related to the metastability of the initially formed structures. A still open question is the nucleation of these structures which usually involves extremely large nucleation barriers and indicates the existence of precursor states not observed so far. Heterogeneous precipitation at extended defects has been summarized for iron, nickel, and copper impurities in silicon and related to the heterogeneous nucleation of precipitate structures that realize large growth rates. The challenging problem of relating the atomic and electronic structures of silicide precipitates has been tackled and has led to initial results concerning the introduction of bandlike and localized states. Theoretical calculations of the electronic structure of silicide precipitates are clearly needed to advance this exciting field. The fundamental knowledge of thermodynamic and transport properties of transition elements was applied to the problem of gettering, Le., the question of how these impunties can be located away from the device-active area to improve the device properties. Gettering techniques were classified into relaxation, segregation, and injection gettering, according to the different mechanism by which they are governed. For interstitially dissolved 3d transition elements, relaxation gettering is dominant for internal as well as for various types of external gettering techniques. Other types of gettering mechanism have been identified for phosphorus diffusion gettering. A quantitative model of PDG, comprising segregation and injection gettering, has been developed recently for Au in Si and has yielded excellent agreement with experimental data. We consider a phenomenological classification as a prerequisite for microscopic models.

The development of such models will allow the gettering efficiency to be evaluated especially with respect to processing temperature and time. A challenge for the future will be the treatment of the simultaneous action of internal and external gettering in silicon materials for solar cells.

Acknowledgements The authors would like to thank Prof. P. Haasen, Dr. H. Hedemann, Dr. K. Graff, Dr. A. Koch, and Dr. E Riedel for their critical comments on this chapter, Profs. G. Borchardt and H. Feichtinger for helpful remarks concerning Sec. 10.2 and K. Heisig for preparing part of the drawings. Financial support by the Sonderforschungsbereich 345 and the German Ministry for Education and Research is gratefully acknowledged.

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11 Fundamental Aspects of Sic

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.

Wolfgang J Choyke and Robert P Devaty Department of Physics and Astronomy. University of Pittsburgh. Pittsburgh. U.S.A.

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 11.1 Polytypism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 11.2 1 1.2.1 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Inequivalent Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.2 666 11.2.3 Some Properties of Simple Polytypes . . . . . . . . . . . . . . . . . . . 667 668 11.2.4 Origin of Polytypism . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 11.3 1 1.3.1 The General Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 11.3.2 The Conduction Band Edges . . . . . . . . . . . . . . . . . . . . . . . . 672 1 1.3.3 The Valence Band Edges . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 11.4 11.4.1 Calculations of Phonon Dispersion Relations . . . . . . . . . . . . . . . 679 I 1.4.2 Infrared Transmission and Reflection . . . . . . . . . . . . . . . . . . . 679 1 1.4.3 Phonon Frequencies Measured by Low Temperature Photoluminescence (LTPL) and the k-space Locations of Conduction Band Minima . . . . . . 681 11.4.4 First and Second Order Raman Scattering . . . . . . . . . . . . . . . . . 682 11.4.5 Raman Scattering from Free Carriers . . . . . . . . . . . . . . . . . . . . 683 Intrinsic Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 11.5 11.6 Shallow Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 11.6.1 Shallow Donors: Nitrogen and Phosphorus . . . . . . . . . . . . . . . . 688 1 1.6.1.1 Nitrogen 11.6.1.2 Phosphorus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 692 11.6.2 Acceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 11.7 1 1.7.1 Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 695 11.7.1.1 Titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 11.7.1.2 Vanadium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1.3 Chromium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 698 11.7.1.4 Molybdenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1.5 Scandium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 699 11.7.1.6 Manganese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Rare Earths: Erbium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 700 11.7.3 Intrinsic Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.7.3.1 Deep Level Transient Spectroscopy (DLTS) . . . . . . . . . . . . . . . . 700 11.7.3.2 Electron Spin Resonance and Positron Annihilation . . . . . . . . . . . . 701

662 11.7.3.3 11.8 11.8.1 11 -8.2 11.8.3 I 1.8.4 11.9 11.10

11 Fundamental Aspects of SIC

Low Temperature Photoluminescence . . . . . . . . . . . . . . . . . . Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . Carrier Effective Masses . . . . . . . . . . . . . . . . . . . . . . . . . . Mobilities and Mobility Anisotropy . . . . . . . . . . . . . . . . . . . . Hall Scattering Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Resolved Measurements and Lifetimes . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 702 704 704 704 705 . 706 708 708


List of Symbols and Abbreviations a aH

B C Cn CN

D (E) E e ED EGX

EX Fvib

g

tz i IC

j Jn

k kB

m m N

n a,

4 RH

rH

S

T t V VT X

a Y3

ES

e

PD P"

lattice constant Bohr radius magnetic flux density Capacitance-Voltage (C-V) stacking sequence lattice constant density of state energy fundamental charge ionization energy indirect exciton energy gap exciton binding energy vibrational contribution to the free energy g-factor Planck constant divided by 2 JG index current of ballistic electrons electric current density coefficient wavevector Boltzmann constant mass mass tensor number of atomic layers number; carrier density Mott criterion wave vector Hall coefficient Hall scattering factor spin temperature time Capacitance-Voltage (C-V) tunnel voltage position absorption coefficient coefficient for nonradiative Auger recombination crystal field splitting static dielectric constant propagation angle drift mobility mobility

663

664


effective “spin” photon frequency ANNNI BEEM CB CVD DAP DDLTS DFT DLTS ENDOR ESR FB f.c.c. FTIR h.c.p. HPT LA LO LTPL MCDA OCDR ODMR PAS STM TA TO

uv VB

axial (or anisotropic) next nearest neighbor ballistic electron emission microscopy conduction band chemical vapor deposition donor-acceptor pair double-correlated deep level transient spectroscopy density functional theory deep level transient spectroscopy electron nuclear double resonance electron spin resonance free-to-bound face centered cubic Fourier transform infrared hexagonal close packed hybrid pseudo potential Longitudinal acoustic Longitudinal optic low temperature photoluminescence magnetic circular dichroism absorption optically detected cyclotron resonance optically detected magnetic resonance positron annihilation spectroscopy scanning tunneling microscopy transverse acoustic transverse optic ultraviolet valence band

11.2 Polytypism

11.1 Introduction The fact that Sic is a newcomer to Vol. 4 of this Series is somewhat surprising considering that its history goes back to before the creation of the solar system. Grains of S i c from outside the solar system have managed to survive and reach the Earth. Astrophysicists now believe (Bernatowicz and Walker, 1997) that this Sic, which is more than 4.6 billion years old (the age of the solar system), originated primarily around stars on the asymptotic giant branch and that a small fraction comes from ejecta of supernova. The concept of a bond between carbon and silicon was suggested as early as 1824 by Berzelius. In 1891, Acheson published the results of his industrial process for making Sic, and in an appendix to Acheson’s paper Frazier gave evidence for the polytypism to be found in the crystals grown by Acheson. Electroluminescence was first seen in S i c in 1907 (Round, 1907). Careful X-ray structural analysis was done by Burdeck and Owen as early as 1918. Rectifying properties and n- and p-type doping of S i c were all established in the 1950s. Its excellent high temperature semiconducting characteristics were also recognized in the 1950s (O’Connor and Smiltens, 1960). So why has it taken until the end of the twentieth century to hit the mainstream of the semiconductor literature? It is only now that relatively large boule crystals [4 in (100 mm) diameter] of a single polytype have become a reality. Uniform epitaxial growth on such a 3 in wafer cut from such a boule has very recently been demonstrated. Doped n- and p-type 2 in (50 mm) wafers are now (1999) commercially available. The dynamic doping range of epitaxial films grown on wafers can range from a high of 1019cm-3 to a low of 1013~ m - With ~ . such S i c available in the 4H and 6H polytype forms, a large number of exciting power and

665

high frequency device structures are possible, and many have been demonstrated. The giant semiconductor market is awakening to these possibilities and consequently the move of S i c into the mainstream. However, S i c cannot be viewed as a simple extension of the well known column IV semiconductors germanium, silicon, or an average of diamond and silicon. Hence the unusual fundamental aspects of S i c are the subjects of this chapter.

11.2 Polytypism 11.2.1 Crystallography As already mentioned, Frazier in 1893 deduced by optical goniometry that crystals of Sic grown by Acheson had numerous crystallographic structures called polytypes. Crystallographers have had a field day with Sic, and have discovered over 170 polytypes to date. Polytypism radically influences the properties of Sic, and thus we will now describe it in some detail. Let us consider a number of possible ways of arranging hard spheres in close packing as shown in Fig. 11-1. For our purposes, let us designate a Si-C atom pair in an A plane double layer as Aa, in a B plane as Bb, and in a C plane as Cc. It is now possible to generate a series of double layer stacking sequences along the principal crystal axis (the z-axis, which is orthogonal to the x- and y-axes shown in Fig. 11-1). For a stacking AaBbCcAaBbCc ... we generate the cubic polytype of 3C S i c or the zincblende form of Sic. If we stack the double layers as AaBbAaBb .,., we generate the hexagonal polytype 2H S i c or the wurtzite form of Sic. Other common polytypes of S i c may be generated by the following stacking sequences of the bi-layers along the z-axis: 4H S i c (AaBbAaCcAaBbAaCc ...),

666


11.2.2 Inequivalent Sites Layer

-... ‘A

.i.

A ..,:sSii ? ,

..+-

B

H

C . ,

a One Atom

AB

~

TwoAtoms Wurtzitc AaBbCcAa Zincblen

j AaBbAaBb

hcp

ABC fcc Others mixed

Sequences

Close-packed planes Figure 11-1. Close-packed planes perpendicular to thec-axis [OOOI](the principal axis) in hexagonal and rhombohedral lattices or the [ I 1 I ] direction in zincblende lattices. The :axis mentioned in the text is perpendicular to the x - J plane.

6H S i c (AaBbCcAaCcBbAaBbCcAaCcBb . . .), and 15R Sic (AaBbCcAaCcBbCc-

AaBbAaCcAaBbCcBbAaBbCcAaCcBbCcAaBbAaCcAaBbCcBb , . .). The notation, called Ramsdell notation, is straightforward: C for cubic, H for hexagonal, and R for rhombohedral. For simplicity in Sic, A, B, C stands for Aa, Bb, Cc. As can be seen, the unit cell repeat distances along the zaxis can get to be very long. In fact one polytype has been identified with a repeat distance along the principal c-axis (z-axis of Fig. 1 1 - 1 ) of 1200 A (120 nm). The question of polytype stability is still a puzzle and is being studied by a number of theoretical groups.

One reason that the long slender unit cells of the various polytypes of S i c are more than acrystallographic curiosity is that these different unit cells have different numbers of inequivalent silicon or carbon sublattice sites upon which to substitute dopants, or about which point defects may form. This feature and the variation of band gap with polytype (to be discussed later) have a profound effect on the properties of the different polytypes and create a family of semiconductors out of Sic. Consider the number of inequivalent sites in five of the simplest S i c polytypes: Zincblende, 3C Sic, and wurtzite 2H have just one site, but 4H S i c has two, 6H S i c has three, and 15R S i c has five inequivalent sites. What does this mean? Nitrogen acts as a shallow donor in S i c when it substitutes on the carbon sublattice, and similarly aluminum acts as an acceptor when it substitutes on the silicon sublattice. Then, in 4H S i c there are two donors/acceptors, in 6H S i c there are three donordacceptors, and in 15R S i c there are five donors and acceptors, respectively, all due to the substitution of a single chemical dopant. The reason for this can be seen more clearly in Fig. 11-2, where we show a schematic representation of the atomic arrangements of the silicon and carbon atoms in the (1 120) plane of the 6H S i c hexagonal pyramid. Generally, if the stacking of the atomic bi-layers of any S i c polytype is represented in the (1 120) plane, then the complicated stacking sequences can be expressed in terms of simple “zig-zags”. In Fig. 11-2, it can be seen that for 6H S i c there is a “zig” of three lattice positions ABCA to the right and a “zag” of three lattice positions to the left ACBA. As shorthand notation, we write (33) for 6H, (22) for 4H, and (232323) for 15R, etc. Again on Fig. 11-2, we depict car-

T 1:

~

7

11.2 Polytypism

6H Sic Inequivalent Sites

12

S. = 15.lA

"like" Planes

0

12

16 16

667

Figure 11-2. Schematic diagram to illustrate the inequivalent sites in 6H SIC. N,, N,, and N3 are nitrogen atoms substituting on three inequivalent carbon sites in the S i c lattice. N, is substituting on a hexagonal site h and N2 and N, are substituting on two quasi-cubic sites k, and k,. The inequivalence of the sites is illustrated, in the table, by the distance of each of the substitutional nitrogen atoms to carbon "like" and silicon "like" planes. The listed distances are measured in units specified by the scale given on the left. The length of the unit cell of 6H S i c is 15.1 A (1.51 nm), as indicated on the left of the figure.

C A B C A B

bon atoms in the bi-layers with small black circles and silicon atoms with larger shaded circles. A bi-layer designated h is one in which the carbon and silicon atoms find themselves in a quasi-hexagonal stacking environment with respect to the neighboring bi-layers. In the same spirit, k, and k2 are bi-layers where the carbon and silicon atoms find themselves in a quasi-cubic stacking environment with respect to their neighboring bi-layers. Let us now substitute nitrogen atoms at the h, k,, and k2 carbon sites and call them N,, N2, and N3, respectively. On the right in Fig. 11-2 is a table which illustrates that distances in the same crystallographic column (A, B, or C) of N,, N,, and N, atoms from carbon and silicon

"like" planes are the same for nearest neighbors but different for second and third neighbors. The nitrogen atoms substituting on the kland k2 sites sense different environments than a nitrogen atom substituting at an h site. This leads to the unusual situation of three distinct nitrogen donors in 6H Sic.

11.2.3 Some Properties of Simple Polytypes The stacking sequences and selected physical properties of 3C Sic, 15R Sic, 6H Sic, 4H Sic, and 2H Sic, the polytypes of greatest interest in current research and development, are given on Fig. 11-3. The lattice constant in the zincblende 3C S i c

668

11

Fundamental Aspects of SIC

Figure 11-3. A summary of some of the physical properties of 3C, 15R,6H, 4H, and 2H S i c . If these polytypes are represented in the ( I 120) plane of a hexagonal pyramid, then the repeat distance along the c-axis would be C,.

modification and the width of the pyramid faces and the lengths of the unit cells parallel to the principal or c-axis (z-axis of Fig. 1 1- 1) are obtained from X-ray measurements. Recently, Bauer et al.’s (1998) very precise X-ray measurements yielded highprecision determinations of atomic positions in 4H S i c and 6H Sic. These measurements were also compared to results from calculations based on density functional theory (DFT) within the local-density approximation. The unit cell geometries of the polytypes dictate the atoms per unit cell and the number of inequivalent sites contained in a particular unit cell. The space groups for the various polytypes are a consequence of the placement of the carbon and silicon atoms in each polytype. The indirect exci-

ton energy gap E,, (eV) , measured at 2 K, is obtained from optical measurements and given at the bottom of Fig. 11-3. The band structure and energy gaps in S i c will be discussed in some detail in Sec. 1 1.3.

11.2.4 Origin of Polytypism The underlying causes of the multitudinous stacking arrangements exhibited by S i c polytypes are not well understood. Explanations based on both high temperature thermodynamic equilibrium states and details of the kinetics during growth have been proposed. One of the earliest proposed growth mechanisms is Frank’s model (1951) of spiral growth around screw dislocations. In the last decade, powerful computational

1 1.2 Polytypism

techniques have been applied to the calculation of bulk cohesive energies of the important S i c polytypes, beginning with the work of Heine et al. (1992a, b). These calculations are challenging because the differences in cohesive energy are only a few millielectronvolts per atom, so that high accuracy is required. Not surprisingly, the results obtained by various groups differ due to differences in their assumptions and procedures. The recent paper by Limpijumnong and Lambrecht (1998) provides a comparison of results and numerous references to the literature. According to calculations, 4H, 6H, and 15R S i c have the lowest total energies of the important polytypes, but which polytype has the lowest energy differs among the calculations. The total energy for 3C S i c is larger, and the value for 2H Sic is the highest. The results of these sophisticated electronic structure calculations are frequently interpreted and compared using the axial (or anisotropic) next nearest neighbor king (ANNNI) model, which was originally developed to explain the appearance of many complex phases, somewhat analogous to polytypes, in certain magnetic materials. For application to Sic, each double layer in the stacking sequence is assigned a value of q = + l or -1 for its ‘spin’, depending on its relationship to the preceding bi-layer in the sequence. Parallel (anti-parallel) ‘spins’ correspond to cubic (hexagonal) stacking. Coefficients J,, in the expression for the energy

describe the interactions between adjacent double layers ( n= l), next nearest double layers (n=2), etc. The index i is summed over the total number of bilayers N . A positive value for a coefficient J,, favors the

669

same stacking configuration for the nth nearest double layers. Typically, the interactions are cut off at n = 3, although higher values of n as well as terms describing interactions among four double layers are sometimes included. At zero temperature, the phase diagram for this model is easily worked out, and there is a point at which an infinite number of phases (polytypes) can coexist ( J 1=-2 J z , J,=O with J l > O ) . According to the calculations of Cheng et a]. ( 1 988), S i c lies very close to this point, suggesting a favorable situation for polytypism. Results calculated by other groups (Limpijumnong and Lambrecht, 1998; Bechstedt et al., 1997), are not so close to this point. Limpijumnong and Lambrecht (1998) use the ANNNI model as a parameterization scheme, using calculated results for enough polytypes to uniquely set the coefficients, then attempt to predict the free energies of additional polytypes. The origin of the interactions between double layers separated by quite considerable distances is not understood. Heine et al. (1992a) discuss an analogy with Friedel oscillations in metals. Recently, Bauer et al. (1998) have carefully reexamined the locations of atoms within the unit cells of S i c polytypes, which may provide clues regarding the relationship between relaxation of atomic positions and stability. There may also be a connection between stoichiometry and polytypism (Tairov and Tsvetkov, 1984). Because the calculated total energies of polytypes are so close, additional small effects may be important. The vibrational con-

+kgTln 1-eXp

670

1 1 Fundamental Aspects of SIC

called the phonon free energy, will differ for each polytype due to small differences in the phonon dispersion relations. Bechstedt et al. (1997) calculated differences about a factor of ten larger than Cheng et al. ( 1 990). There is also qualitative disagreement: Cheng and co-workers find that the phonon free energy tends to stabilize 6H Sic with respect to 4H Sic with increasing temperature, while Bechstedt and co-workers find the opposite. Some papers have attempted to relate results of bulk calculations to the mechanism of growth. For example, the parameters of the ANNNI model can be used to describe the addition of a layer (Heine et al., 1992b), but it is highly questionable whether the Values of these parameters obtained from bulk calculation apply, even if the ANNNI model is valid. Perhaps a more promising avenue for progress in the near future is the detailed study of growth mechanisms using modern methods of surface science.

11.3 Band Structure 11.3.1 The General Picture A fairly thorough review of theoretical topics relating to fundamental properties of Sic up to early 1997 is given by Choyke et al. (1997). Most of the theoretical discussions are related to the band structures of Sic polytypes, and the progress that has been made on band theory since the current upsurge of industrial interest in S i c in this decade is impressive. During this period, much improved size, polytype, and dopant control of bulk crystals of Sic has been obtained, making possible excellent epitaxial growth on wafers cut from such bulk crystals. This in turn has made possible experimental verification of the theory, and points to areas that require much further study. To give a general overview of the band struc-

ture of 2H, 4H, 6H, and 3C S i c we show in Fig. 1 1-4 a slightly simplified version of the electronic band structures E ( k ) calculated by Chen and Srichaikul (1997). The positions of the valence band maxima and the conduction band minima are indicated for these four polytypes. From this, it follows that these S i c polytypes are large band-gap indirect semiconductors. All S i c polytypes measured to date are indirect semiconductors and we expect all S i c polytypes to follow this pattern. To the right of each of the four calculated polytype band structures in Fig. 11-4 we give the experimentally obtained values of the exciton band gaps EGx. These authors use a hybrid pseudo-potential (HPT) and tight binding approach to obtain the band structures in Fig. 11-4. We chose these calculations to illustrate the band structures of 2H, 4H, 6H, and 3C Sic, because the band-gaps are fitted to the given experimental values. A number of excellent first principles calculations have also become available recently (Choyke et al., 1997; Persson and Lindefelt, 1997), and there is good agreement among these calculations. In Fig. 11-4 we have arranged the polytypes with the 2H wurtzite structure at the top and the 3C zincblende structure at the bottom. This ordering in terms of “hexagonality” shows a number of interesting trends. As we go from 2H S i c to 3C S i c the forbidden gap becomes progressively smaller. Lambrecht et al. (1997) have shown how this comes to pass from their band theoretical considerations. It is important to devise experimental tests for the determination of the qualitative reliability of the new band structure calculations. Optical reflectivity measurements on various polytypes in the middle ultraviolet region (4-1 1 eV) on large “as-grown” optical surfaces are now possible due to improved substrate wafers and epitaxial growth. Such reflectometer measurements combined

11.3 Band Structure

5

t ;=^ w v

2H Sic E,,

0

m~

r

M

L

A

(2K) = 3.330 e~

671

Figure 11-4. The bandstructure of 2H, 4H,6H,and 3C Sic near the valence band maximum (VB,,,) and the conduction band minimum (CBmin); EGX ( 2 K) is the exciton band gap in elecuonvolts measured at 2 K.

H K

5

t

4H Sic

2

eo w

EGX (2K) = 3.265 eV

r

M L

A

HK

5

t

6H Sic EGX(2K) = 3.023 eV

3C Sic EGX (2K) = 2.390 eV

with a strong theoretical effort (Lambrecht et a]., 1993, 1994; Suttropet al., 1993) have given a detailed interpretation of the measured spectra and provide a strong test of the theory. Room temperature spectra were reported for 4H, 15R, 6H, and 3C Sic, while

the reflectivity was calculated for 2H, 4H, 6H, and 3C Sic. This allowed a consideration of trends among polytypes. While many features in the measured reflectivity may be interpreted using calculated critical point transitions at symmetry points in the Bril-

672

11 Fundamental AsDects of

SIC

louin zone, an important result of this work is that the major peaks in the reflectivity are associated with rather extended regions of k-space, over which the energy difference between two bands is nearly constant. Calculations of the UV optical properties were extended to 25 eV by Lambrecht et al. ( 1 997).

11.3.2 The Conduction Band Edges Of particular value for the modeling of S i c devices is the guide that band theory gives us in locating the positions of the conduction band minima in the 3C S i c zincblende Brillouin zone, the 2H, 4H,and 6H Sic hexagonal Brillouin zones, and the 15R Sic rhombohedral Brillouin zone. Various theoretical groups, applying slightly differ-

t

ent approaches, have obtained the effective mass tensor components for the bottoms of the conduction bands of these polytypes. Experiment has met with major obstacles in accurately confirming the theoretical effective mass tensor values, except in 3C S i c and 4H Sic. In Fig. 11-5 we have adapted a table from an article by Wellenhofer and Rossler (1997a) to summarize the current theoretical and experimental findings at the conduction band minima for 2H, 4H, 6H, 15R, and 3C Sic. To simplify the figure we have left out references to the various calculations and measurements, but these may be found in the paper of Wellenhofer and Rossler (1 997 a). Lambrecht et al. (1997) gave a theoretical interpretation, based on the band struc-

k.

3C Sic m mA

minimum at 0.70 0.23

2H Sic

0.63 0.23

0.68 0.23

0 40 0.26

0.45 0.26

0.45 0.27

0.43 0.26

mu-r

0.66

0.31 0.30

0.57 0.32 0.32

0.58 0 28 0.31

0.57

mM.h

mu.L

mu-h mM.1

0.28 0.31

minimum along M-L

6H SIC mM.r

0.60 0.29

0.667 0.247

0.67 0.25

0.67 0.22

0.58 0.31 0.33

ml:0.18 0.30

minimum at K

ml,

ml

k

0.67 0.25

X

0.78 0.23 12-2.0

15R SIC mx r

0 67

mX h

0 22

mx u

041

0.75 0.27 1.95

0.77 0.24 1.24

0.75 0.24 1.83

(I

m,,:0.22

0.48

m l : 0.35

0.24

0.25 0.42

ml,: 1.4

0.34

1.7

mL 0 2 8

024

053

038

2.0

minimum at X

rnll

Figure 11-5. Theoretical and experimental values for the effective electron masses in 3C, 2H, 4H, 6H,and 15R Sic. The columns compare theoretical as well as experimental values obtained in a variety of ways.

11.3 Band Structure

tures of undoped S i c polytypes, of the optical absorption attributed to the first few inter-conduction band transitions in heavily n-type doped Sic. This optical absorption has been evident in heavily n-type doped platelets since Acheson’s time, leading to a variety of colorful platelets for different polytypes. Measurements on such platelets were first reported in 1965 by Biedermann on 4H, 6H, 15R, and 8H Sic, and on 3C S i c by Patrick and Choyke in 1969. At the time, it was supposed that the observed absorption bands were due to transitions from the lowest conduction band to higher conduction bands. Our current understanding requires a generalization in that the modern band structure calculations indicate that in polytypes such as 4H S i c there is a second conduction band very close to the lowest conduction band (1 30 meV), and hence under heavy doping the second band may be

673

partially occupied and participate in the optical absorption. We shall now illustrate details of this interband absorption for polytypes 4H and 6H. For 4H Sic, it appears adequate to consider transitions in the Brillouin zone at the conduction band minimum, which has been located to be at point M. In the case of 6H Sic, however, we must take into account the entire M-U-L axis, due to the fact that the lowest conduction band is extremely flat in this direction. In Figs. 11-6 and 11-7 we show the theoretical band structure interpretation of the optical absorption for 4H and 6H Sic, as well as the experimental optical absorption data. The figures are based on Fig. 11-1 in Lambrecht et al.’s paper (1997). In contrast to the valence to conduction band indirect transitions where phonons play such a vital role, we show only direct transitions in Fig. 1 1 -6 for the inter-conduction

4H SiC:N EXPERIMENT

FI 150

,

I 100

,

THEORY

I-jO.0 50

c--a (cm-’) Absorption Coefficient

Energy Levels at M point for CB Minima

Figure 11-6. Comparison of experiment and theory for inter-conduction-band optical transitions in heavily doped n-type 4H Sic.

674


Figure 11-7. Comparison of experiment and theory for inter-conduction-band optical transitions in heavily doped n-type 6H SIC.

6H SiC:N THEORY

EXPERIMENT 3.0

CB6 CB5

2.5

2.0

I .5

m

U

P

1 .o

8

2

0

0.5

Y

0.0 150

100

50

+---a (cm-’)Absorption Coefficient

M

U Wavevector

band transitions in 4H S i c near the M point. Symmetry arguments are used to assert that the inter-conduction band optical transitions are allowed between bands of the same symmetry for E // c and allowed for bands of different symmetry forE Ic . Agreement between experiment and theory is very satisfactory. In the case of 6H S i c (Fig. 1 1-7), transitions from CB I to bands up to CB, are considered. In 6H Sic, the whole M-U-L axis is believed to be important in explaining these transitions. Again the selection rules and the calculated band structures reproduce the observed absorption peaks reasonably well, except for the separation of the peaks labeled A and B. Lambrecht et al. (1997) believe that the A-B fine structure can be explained in terms of the double minimum (Camel’s Back) structure of the lowest 6H S i c conduction band. Recently, ballistic-electron emission microscopy (BEEM) has been used to get

L

k ---t

quantitative values of the energy separations of several of the lowest conduction bands near the minimum. Im et ai. (1998a, b, c) and Kaczer et al. (1 998) have reported BEEM measurements on platinum and palladium Schottky contacts in 4H and 6H Sic, as well as palladium Schottky contacts in 15R Sic. In the BEEM technique, the dependence of the current of ballistic electrons (I,) from the scanning tunneling microscope (STM) tip into the Sic sample as a function of tunnel voltage (V,) at constant tunnel current, can be measured. In this way, the conduction band structure of S i c may be probed. Figure 11-8 gives BEEMdatafor 6H and 4H S i c for both palladium and platinum contact films, but for 15R Sic, only data for palladium contact films is shown. We show in Fig. 11-8 plots of f, (PA) as a function of V,. Each set of curves represents an average of more than one hundred individual Zc-VT data sets. In the cases where

11.3 Band Structure

1.o

1.5

2.0

4

2.5

I

3

u

2

a

v

U

U

1

0 I 1.o

1.o

0.5

1

I .5

2.0

2.5

675

both the palladium and the platinum curves are shown, the platinum data is offset by a constant factor from the palladium curve for the sake of clarity. Bell and Kaiser (1988) have given a theoretical model for the BEEM spectrum, which has been used to fit the data near the threshold, giving rise to the solid curves. BEEM can also detect higher lying conduction band minima when the STM tip voltage (V,) reaches a value such that the hot ballistic electrons have enough energy to reach a higher conduction band minimum. An onset of additional BEEM current (I,) is then expected, which can best be observed by taking a derivative of the BEEM (Zc- V,) curve. Such derivative curves are shown as insets on the three data plots of Fig. 11-8. The single threshold in 6H Sic and the first and second thresholds in 4H and 15R S i c are indicated by arrows pointing to the changes in slopes of the derivative curves in the insets. The arrows on the integral curves indicate the threshold voltages determined from the fits to the Bell-Kaiser model. In the case of 6H Sic, Schottky barrier heights of 1.272 0.02 eV for palladium and 1.34 20.02 eV for platinum are obtained. For 4H Sic, the palladium threshold is 1S4k0.03 eV and the platinum threshold is 1.58k0.03 eV. Note that the difference of the Schottky barrier energies between 4H and 6H S i c is 0.24 eV, which is just about the difference between the measured exciton band gaps for these two polytypes. In the case of 15R Sic, the threshold for the

v

4

U

U

0.0

Figure 11-8. Averaged BEEM (1, versus V,) data taken from metal (palladium/platinum) 6H, 4H,and 15R S i c contacts. The platinum curve is vertically offset for clarity. Solid curves are fits to the data using the Bell-Kaiser model with arrows indicating thresholds extracted from the fits. Insets are the derivative spectra dIJdV, as a function of V, for palladium contacts, where the changes in slopes are indicated by arrows.

676


palladium contacts is about 1.22 eV. The difference between the exciton energy gaps of 6H S i c and 15R S i c is 37 meV, whereas the difference in the Schottky barriers is about 50 meV. The difference of 13 meV is well within the error of the measurements. As already mentioned, for 6H Sic we see no additional onset of tunnel current. However, for both 4H and 15R S i c we see second thresholds in the derivative curves. For 4H Sic, a measured energy difference of approximately 140 meV is obtained between the minima of the two lowest conduction bands, CB, and CB,. For 15R Sic, the derivative curve also shows two clear onsets separating CB and CB, by about 500 meV. A comparison of these BEEM results with band calculations is shown in Fig. 1 1-9. The top row displays the lowest lying conduction bands in 6H, 4H, and 15R S i c as obtained from theory. In 6H S i c we see no second conduction band minimum within 0.6 eV of the minimum of CB, . In 4H S i c there is a second higher minimum at the

4H S iC

6H S i c t

M point approximately 130 meV above the first minimum. For 15R S i c theory predicts two minima, one at point X of CB, and the second at the point L of CB, in the rhombohedral Brillouin zone of 15R Sic, approximately 500 meV above the minimum at X. The derivative BEEM spectra shown in Fig. 1 1-8 do not directly map the density of states of the S i c polytype, but it might be expected that each higher lying conduction band minimum will introduce an abrupt increase in the density of states. Consequently, in the second row of Fig. 11-9 we show the theoretically calculated curves for the density of states of 6H, 4H, and 15R S i c as a function of energy above the zero of energy at the minima of the lowest lying conduction bands. The arrows on the density of states curves indicate the onset of an increase in the density of states. For 6H S i c no additional increase in the density of states is seen, but for 4H S i c we see one at about 120 meV and for 15R S i c there is one indicated at roughly 500 meV.

0.6

15R S i c

0.6

0.6

';

0.4 0.2 0.0

M

M U L

M

L

L

CBZ

? I

-

4

%- 2 EO

=El

'

1

X

A

CBt

0 00

02

04

E (e V)

06

--+

08

00

02

04

E (eV) --+

06

011

00

02

04

06

011

E (e V) --+

Figure 11-9. Calculated conduction bands and density of states for 6H, 4H, and 15R S i c near the conduction band edges (Im et al., 1998; Kaczer et ai., 1998).

11.3 Band Structure

677

11.3.3 The Valence Band Edges 0

Considerably less is known about the valence band edges of S i c than about the conduction band edges. However, it is generally agreed that the maximum of the valence bands in all polytypes is at or very near the rpoint in the Brillouin zone. In zincblende or 3C S i c the top of the valence band is sixfold degenerate, including spin but neglecting the spin-orbit interaction. The spinorbit splitting for 3C S i c has been experimentally determined to be about 10 meV (Humphreys et al., 1981). If we include the spin-orbit interaction, we have fourfold and twofold degenerate bands at the rpoint. For the hexagonal and rhombohedral polytypes, we must also take into account the effect of the crystal field. In 6H, 15R, 4H, and 2H S i c the crystal field splitting is expected to be considerably larger than the spin-orbit splitting. If we include the crystal field splitting for these polytypes, we obtain three twofold degenerate valence bands at the rpoint. A number of theory groups (Persson and Lindefelt, 1996, 1997; Lambrecht et al., 1997; Wellenhofer and Rossler, 1997 a) have recently made calculations of the details of the valence band structure near the rpoint and have given estimates of the hole effective masses for the three highest lying valence bands. In Fig. 11- 10 we show the valence band edge of 6H S i c (Wellenhofer and Rossler, 1997 a). The spin-orbit splitting has been adjusted to agree with the measured spinorbit splittings reported by Humphreys et al. (1981). The crystal-field splitting is also in good agreement with experimental findings, and is discussed later. Lambrecht et al. (1997) among others have given a theoretical treatment in which they show that the crystal-field splitting (the region ACF 10 pm min-'), and the material yield is high since substrate and source are close together. Another deposition process which is used for production is chemical spraying. On a small commercial scale for consumer applications, CdTe films are at present deposited by screen-printing. There are some doubts on the feasibility of this process for large-scale production. The alternatives are to deposit the CdTe layer by the same techniques as the CdS layer. In addition, electrodeposition is a viable option. Although slow, it can be applied to a large

757

number of substrates in parallel. The deposition occurs either at temperatures above 500 "C or the films are annealed later. Contacting the p-type CdTe layer on the back side is a difficult task. Basically, there are two general principles for making ohmic contacts to p-type semiconductors: 1) Deposition of a metal that has a work function higher than the electron affinity of the semiconductor in order to align the upper valence band edge to the Fermi level of the metal. 2) Formation of a highly doped surface layer thin enough for holes to tunnel through to a metal contact.

In both cases there are problems for CdTe. Low cost metals with appropriate work functions > 5 eV are not available. P-doping by diffusion from the surface has the drawback that dopants generally diffuse preferentially along grain boundaries, leading to shunting before a sufficient doping level can be achieved. Using thicker than necessary films can alleviate the latter problem, but in practice another approach is pursued. The CdTe film is combined with another semiconductor, which is easier to dope and then contacted with an appropriate metal. The highest efficiency cells have been obtained with copper-doped graphite. Another possibility is copper-doped ZnTe. In both cases, the danger exists that upon annealing copper can rapidly diffuse into the CdTe film and reduce the performance. Two other p-type semiconductors have been considered, namely HgTe and tellurium, but in all cases the modifications of the surface remain a delicate process, which depends on the material and its microstructure. 12.5.1.2 Electronic Properties The performance of the solar cell depends very much on the quality of the interface re-

758

12 New Materials: Semiconductors for Solar Cells

gion between CdS and CdTe. The CdS has a significant lattice mismatch to CdTe, which causes a high density of interface defects and states. The problem is however less severe, since it has been observed that a mixed CdS,TeI-,2 phase is formed at the interface. The performance of the solar cell, i n particular the open circuit voltage, depends on the formation of an appropriate interface phase: best intermediate layers achieve 850 meV, whereas for "insufficient" mixing the voltage stays below 800 meV. It is assumed that formation of the intermediate phase is responsible for the reduction of structural defects at the interface and thereby of recombination centers (Jensen et al., 1996: Al-Ani et al.. 1993). The mixing is promoted by three factors: (1) a smaller grain size of the CdS film, (2) a higher deposition temperature of the substrate, and (3) in-diffusion of C1 ions. The formation of smaller grains in the CdS film requires a low temperature deposition process, such as chemical bath deposition at 70 "C, whereas closed spaced sublimation at 400-500 "C is less favorable. In practice, only relatively thin CdS,Te,-, layers can be formed, which limits the performance. In the thin film technology, particularly of compound semiconductors, maximum Performance can often only be achieved by special treatments after deposition. These procedures can be heat treatments, annealing in a certain gas atmosphere, or the indiffusion of certain elements, such as hydrogen. Although in many cases the physical processes are not completely understood, it can i n general be assumed that both microstructural changes and/or the removal of electrically active defect states occur. For the CdTeKdS device, it has become common practice to diffuse C1 ions into the film from a CdClz source which has been deposited on the surface. In other cases, such as screen printing or electrodeposition, the C1

ions are supplied during the growth process of the CdTe layer. Detailed investigations show that mainly the density of CdTe/CdS interface states is reduced by the activation process. Unfortunately, at the same time new defect centers are formed in the space charge region, which limits the possible improvement (Bonnet, 1997; Rohatgi, 1992). On small cells using fabrication processes and materials close to production conditions, efficiencies of 14.3% have been obtained, and for laboratory cells 16%. Recent industrial production results for modules are somewhat lower at around 9-10%. The advantage of the technology is the reduced cost compared to that of silicon cells. There are, however, concerns about environmental and health problems with the widespread use and deployment of cadmium. Although incineration studies have shown that the emission of toxic cadmium during the exposure of modules to fire is negligible, doubts remain as to whether large scale cadmium technology is acceptable.

12.5.2 Chalcopyrite Semiconductors 12.5.2.1 General Properties of CuInSe,! and Related Compounds Ternary chalcopyrite semiconductors with the composition A' B"' X:' have attracted interest because of their diverse optical, electrical, and structural properties (Gibart et al., 1980; Kuriyama and Nakamura, 1987). Details of the compounds that have been most thoroughly investigated so far are summarized in Table 12-2. Some have band-gap energies in the range 1-2 eV and are attractive candidates for photovoltaic applications. The research focused mainly on the compound CuInSe, and the related quaternary compound Cu(In,Ga)Se2, but CuGaSe, and CuInS, with even more suitable band-gap energies also open prospects for future thin film devices.

12.5 Polycrystalline Thin Film Compound Semiconductors

Table 12-2. Structural and electronic properties of some copper-based ternary compounds with chalcopyrite structure.

CuGaS2 CuInS, CuGaSe2 CuInSe, CuA1Te2 CuGaTe, CuInTe,

5.35 5.52 5.596 5.782 5.964 5.994 6.197

1.959 2.016 1.966 2.0097 1.975 1.987 2.00

2.43 1.43 1.68 1.04 2.06 1.23 1.04

975 -

810 -

672

u lattice constant; c long axis of the unit cell; E, band gap energy; TYdtransition temperature between ordered and disordered phase. a

The chalcopyrite lattice can be developed from the sphalerite structure by duplicating the unit cell and arranging the cations A and B on the cation sublattice (Fig. 12-29). The unit cell is characterized by a tetragonal dis-

Figure 12-29. Tetragonal unit cell of the chalcopyritelatticeforanABX,compound. Theunitcellischaracterized by a tetragonal distortion along the c-axis. Because of the different bond strengths, the anion X usually adopts an equilibrium position that is closer to one pair of cations than to the other. In addition, the structure of a vacancy-antisite defect pair (2 V , + B A ) is shown, which has been proposed for CuInSe,.

759

tortion along the c-axis (Jaffe and Zunger, 1984; Romeo, 1980). It appears to be acommon feature of many copper-ternaries that a phase transition from the chalcopyrite to the cubic sphalerite structure occurs at elevated temperatures due to a random distribution of A and B atoms in the cation sublattice. It is also a characteristic feature of the chalcopyrites that the structure can be maintained over a rather wide range of compositions, in some cases several percent. It is assumed that this is achieved by the incorporation of intrinsic point defects such as vacancies, interstitial atoms, and antisite defects. Absorption measurements for single- and polycrystalline CuInSe, show an enormously high absorption for this material in comparison with other semiconductors, and confirm the behavior as a direct semiconductor (Kazmerski and Wagner, 1985; Shah and Wernick, 1975; Kazmerski, 1977). Specimens prepared by different techniques and subjected to various annealing treatments show, however, that the empirical band gaps vary between 0.92 and 1.04 eV. It has been suggested and experimentally verified that tail states and a band gap narrowing occur due to high concentrations of intrinsic shallow doping defects (Neumann, 1986), which can be related to the composition and microstructure. The high density of intrinsic point defects determines not only the optical absorption to some extent, but also the conductivity and other electrical properties. For instance, CuInSe, and CuInS, can be made n- or ptype by changing the stoichiometry. Typical values for the carrier concentrations in nand p-type single crystal CuInSe, are in the range 10l6- lo” ~ m - Measurements ~. of the Hall mobilities show, however, that the concentrations of ionized defects can be as high ~ , is considerably higher as 10’’~ r n -which compared to the free-carrier concentration.

760


This indicates a high degree of compensation of acceptor and donor-like defects, and seems to be characteristic for copper-ternary semiconductors. Systematic investigations of the relationship between the conductivity and stoichiometry of single crystals are depicted in Fig. 12-30 for CuInSez (Neumann and Tomlinson, 1990; Noufi et al., 1984). It is now generally accepted that the results can be explained on the basis of a point defect model, which assumes that intrinsic point defects compensate for the deviation from ideal stoichiometry and introduce shallow acceptor and donor levels in the band gap. The intrinsic point defects, which have to be considered in general, are vacancies, interstitials. and antisite defects. In a compound with the general formula ABX?, twelve different native defects have to be considered,

Figure 12-30. Experimental results for the n- and ptype conductivity as a function of the stoichiometry for CuInSe? single crystals and comparison with a numerical calculation using the theoretical formation energies @\.enin Fig. 12-31, The experimentally determined energy levels of the intrinsic defects, which habe been used i n the calculation, are summarized in Table 12-3. In polycrystalline thin films, the p-type region is enlarged (light area) due to influence of the grain boundaries on the band-gap states.

three vacancies: VA,V,, V,, three interstitials: A i , Bi, Xi, and six antisite defects: A,, BA, A,, B,, X,, X,. In addition, the formation of complexes is feasible because the concentrations can become very high. Chalcopyrite semiconductors are mainly covalently bonded compounds but with a high degree of ionicity. Identification of the dominant doping defects for each composition has therefore been based on the assumption that the ternary chalcopyrites can be analyzed in a similar way to ionic crystals. In the Kroger model, the defect chemistry of ionic crystals is determined by certain equilibria between intrinsic defects, which can be described by a mass law of action (Kroger, 1983; Groenink and Janse, 1978). In thermodynamical equilibrium, the defect concentrations are not independent of each other but related by these internal equilibria. The concentrations depend on the stoichiometry of the crystal. For a given composition and temperature, the concentrations of all defects are completely determined by these internal equilibria and can be calculated, provided that their formation enthalpies are known. For CuInSe, the formation energies AHf of some defects have been determined recently by a first-principle, self-consistent, electronic structure calculation (Zhang et al., 1997). An important new aspect is that the formation energies are not fixed constants but vary considerably with the Fermi level and the chemical potential of the atomic species (Fig. 12-31). This means that the formation energies also depend on the composition of the crystal. Since the formation energies of the anion-cation antisites and the selenium and indium interstitial appear to be too high, only six native defects have to be considered. The defect with the lowest energy is the copper vacancy, while some of the other defects, such as the indium vacancy and the antisites CuInand Incu, have


761

Figure 12-31. The formation energies of the main intrinsic defects in CuInSez as a function of the atomic chemical potential of copper (kJ and indium (,uln).Neutral and charged defects are considered. Intermediate values of the chemical potentials are obtained by linear extrapolation between the values at k L C U = p l n(Zhang = - 2 et ai., 1997).

-2.0

-1.5 Cu-poor

-1.0 p~,,

-0.5

0

-0.5

-1.0 pln

-1.5

-2.0

In-poor

Chemical potential [eV]

comparably low, even negative, energies in some composition regimes. Another important result is that the interaction of the copper vacancy and the indium-copper antisite can lead to a complex 2Vcu + Incu, with a lower formation energy compared to the copper vacancy Vcu. The atomic structure of the defect is shown in Fig. 12-29. It is dominant for copper-poor crystals, and because of the donor character should be responsible for the n-type behavior in this regime. Based on these results and experimental information about the electronic levels in the band gap (Table 12-3), the concentrations of all defects, electrons, and holes have been calculated as a function of the composition (Klais et a]., 1998). The results for the conductivity behavior are included in Fig. 12-30 and show rather good experimental

agreement for most of the compositions. A characteristic feature is that in a narrow range of compositions the carrier concentration drops by several orders of magnitude and the conductivity changes from n- to p-type behavior, or vice versa. The results also confirm that a high concentration of compensating acceptor and donor defects occurs. The electronic properties of a polycrystalline compound film, and possibly the defect chemistry as well, may differ considerably from the monocrystalline behavior, because of the additional grain boundary states, as discussed in Sec. 12.4.3. This general behavior is also observed for copperternaries, where the correlation between conductivity and composition for selenium deficient polycrystalline films differs significantly from the single crystal behavior

762


Table 12-3. Calculated energy levels E, of the main intrinsic defectsa and comparison with experimental results (in parentheses). Acceptors

;V :

vo/In

VI-,/?-

vkl3-

cuy;290 (220-320)

EV + ET

30 (30-45)

170 (120-200)

410 (400)

670

- ET

100 (55-90)

250 (200-232)

340 (350-370)

200 (120-232)

cu-/2In 580 (570)

200

a Zhang et al. ( 1997); the theoretical values have been used for the calculation of the defect chemistry and conductivity shown in Fig. 12-30.

(Noufi et al., 1984). It is also verified that the grain size has a significant influence on the lifetime of charge carriers (Bacher et al., 1996). If the composition of the films is near the ideal stoichiometry, the material is always p-type, whereas a large deficiency of copper converts the conductivity into n-type behavior (Fig. 12-30). The band gap of CuInSe, below the optimal value of 1.5 eV for photovoltaic applications, limits the utilization of this material for single-junction devices. Since the band gap increases from CuInSe, (1.04 eV) to CuGaSez (1.68 eV), the formation of a quaternary compound CuIn,Ga,-,Se2 offers the possibility to adjust the band-gap energy by changing the In/Ga ratio. The highest efficiencies of about 15- 1 7 8 for laboratory cells are actually based on the compound CuIn,Ga,-,Se, (CIGS). The development of a CuInSe,-,S,, cell is still hampered by the fact that with increasing sulfur content, the p-type conductivity decreases. It has been shown, however. that small additions of sulfur can increase the efficiency of CIGS cells, which demonstrates the beneficial effect of sulfur. The typical CIS device structure is based on the heterostructure concept with borondoped ZnO as a window material (wide band

gap) on top (Fig. 12-5 b). In practice, a thin intermediate layer of CdS is required, although the role of the buffer is not well understood. In the future it would be desirable to replace the heavy metal in the buffer layer by an alternative compound. Some possible candidates are ZnSe, In,S, (Karg et al., 1997), and indium or tin hydroxy compounds (Hariskos et al., 1996), but these are less effective so far compared to CdS. The films are deposited on soda lime float glass, which yields the highest efficiencies. This is due to the release of sodium into the semiconductor film, which appears to have a beneficial effect on the device’s performance.

12.5.2.2 Deposition Techniques For large scale production, two deposition techniques for the CIS absorber layer are mainly considered: ( 1 ) Selenization of metal precursor layers

and subsequent annealing. For example, for the CIS fabrication, InSe and copper precursors are deposited on the substrate. Selenization occurs by annealing in H,S or selenium vapor or from a selenium precursor layer. Relatively large


grains (>1 pm) develop under these conditions. In this process, the composition of the absorber is limited to indium-rich material. The incorporation of gallium is also difficult and limited to small amounts. (2) The best films are obtained by co-evaporation of all elements. The process gives full flexibility in device optimization, but is more difficult to incorporate in a large scale fabrication process. In general, CIS films grown under copper-rich conditions show better crystalline quality than those grown under indium-rich conditions. For high copper : indium ratios the compensation reduces significantly, but the hole concentration increases above 1019 ~ m - In ~ .practice, indium-rich p-type films are therefore grown, which have a more suitable resistance. Sodium can be incorporated either from the glass substrate or directly during the deposition process. The n-type CdS layer is deposited by a chemical bath process, and is subsequently dried. The transparent conductive ZnO window layer is mostly deposited by sputtering with a bi-layer structure made of an inner undoped layer and outer aluminum- or boron-doped layer. ZnO has a band gap around 3.3 eV and thus has high transmission in the visible range. The main purpose of the window layer is to reduce recombination at the front surface, which is important for a thin film device.

12.5.2.3 Electronic Properties The device’s properties are mainly determined by the electrical properties of the bulk absorber (CIS or CIGS), the buffer material (CdS, etc.), and the interface. Experimental results confirm that a good microscopic crystal quality is an essential basis for a good solar cell. Because of the complex defect chemistry of ternary chalcopyrites in

763

combination with the presence of a high density of grain boundaries in the polycrystalline films, the interpretation of experimental results is, however, a very difficult task. In addition, the role of sodium and oxygen has to be considered, mainly in the CdS layer. It is therefore not surprising that many of the observed phenomena are not fully understood, despite a wealth of information. Since in chalcopyrites large deviations from stoichiometry can occur, the density of intrinsic defects with shallow donor and acceptor character and the compensation may be high. Indeed, it has been observed that the analysis of electrical and optical results has to take into account a high density of donor and acceptor defects. Any inhomogeneous distribution in the crystal can lead to potential fluctuations, which influence the optical and electrical properties. For instance, experimental results on the optical transitions between impurity tail-states near the band edges have been explained by potential fluctuations (Karg et al., 1997). For the ZnO/CdS/CuInSe, system, a considerable increase in efficiency was achieved by the use of soda lime glass as a substrate material. During deposition of the CuInSe,! absorber film, sodium diffuses from the substrate and disperses into the film. It could be shown that sodium affects not only the grain growth and conductivity properties of the film but also the CdS interface formation and properties. There is evidence that it gives rise to a shallow acceptor state that increases the effective carrier concentration of the absorber (Probst et al., 1996). Sodium also reacts with SeO,, which mainly occurs at the surface, and leads to an enrichment of selenium there. It has been verified that the lifetime of the charge carriers in the absorber is significantly influenced by the grain size. Therefore the grain boundary states are important for nonradiative recombination processes.

764


However, the origin of the defect states has not been clearly identified and the role of extrinsic defects, such as sodium, trapped at the grain boundaries cannot be ruled out yet. Whether the observed improvement of the efficiency after sodium diffusion is due to the impact on the grain size and structure or to the impurity levels themselves is unclear. An important electrical property of a heterostructure interface is the change in the energy gap and the alignment of the band edges (band-offset). Investigations of CdS layers deposited on cleaved CuInSe, in UHV have shown that there is an offset of the valence band edge by 0.8 eV and of the conduction band by 0.65 eV (Loher et al., 1995: Schmid et al., 1993). These are values that cannot be operative in a real cell solar, since for the achieved performance a band offset close to zero is expected. This indicates that under normal processing conditions the interface composition and structure must be changed. Although it could be shown that sodium contributes to the band alignment, this effect alone could not explain the beneficial effect of sodium on the efficiency. For technological applications, the longterm stability of the devices and modules is a crucial aspect. Considering the main thin film materials today, each one has its own stability problems. For amorphous silicon it is the light induced defect formation and corresponding degradation of the device, for cadmium telluride it is the ohmic contact to the CdTe, and for CIS or CIGS it is the sensitivity to humidity. Water vapor is known to reduce the power output of a CIS solar cell. Since sodium enriches in the bulk and at the surface when CIS layers are deposited on sodium lime glass, interactions between sodium and water can lead to chemical reactions. It has been suggested that these affect the electrical transport proper-

ties, but the detailed nature of these effects and the role of water vapor are not yet resolved.

12.6 Special Solar Cell Concepts 12.6.1 High Efficiency Solar Cell Materials GaAs and InP are direct semiconductors with an optimum band gap between 1.4 and 1.6 eV (see Fig. 12-3) and a high absorption coefficient. Only thin layers of about a few micrometers are required for a solar cell. Since these materials have also received considerable attention for high speed and opto-electronic applications, a number of deposition techniques are available, such as MOCVD or LPE. The epitaxial growth of layers with a low density of defects requires a monocrystalline substrate of the same material or at least a semiconductor with similar properties. The important parameters are the lattice constants and the band gap energy, which are summarized for some semiconductor compounds in Fig. 12-32. The systems that have to be considered here are InGaP and AlGaAs in combination with GaAs and germanium. However, even for a system with a good lattice match, the defect density of the epitaxial layer is to a large extent determined by the defect density of the substrate material. Experimental results show that, for instance, dislocations from the substrate grow into the epilayer or give rise to the nucleation of other defects. Therefore for high efficiencies it is necessary to start from a substrate crystal with a high material quality, which in most cases is a requirement that at present can only be fulfilled by a few semiconductors, such as GaAs or germanium. Silicon is less suitable because of the difference in lattice constant. The lattice match to GaAs can, however, be

12.6 Special Solar Cell Concepts

4.0

T=300K

3.6 3.2 2.8 Y

w" h F

B

!M ?

.u

0

tt

AIP

0 ZnSe

2.4

2.0

765

Figure 12-32. Energy band gap and lattice constant for some semiconductor elements and compounds. Solid lines connect binary compounds, which are used to form ternary compounds. For ternary compounds with a noncubic chalcopyrite structure, the smaller lattice constant is taken.

1.6 1.2 0.8 0.4 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 htticeconstant a [A]

improved by inserting a strained SiGe buffer layer: Si/Si,Ge,-JGaAs. Epitaxial techniques are therefore in general rather expensive at present and only useful for high efficiency solar cells or if cost considerations are less important, for instance, for application in spacecraft. An advantage of the epitaxial thin film techniques is that they allow fabrication of complex device structures with cells of different band gaps stacked upon each other (tandem cells), which can utilize the spectrum of the light more efficiently. Efficiencies around 30% have been achieved for a number of monolithic tandem structures based on AlGaAs/Si (Umeno et al., 1994), AlGaAs/GaAs (Chung et al., 1989), and InGaP/GaAs (Andreev et al., 1997). 12.6.2 Dye Sensitized Ti02

A new kind of solar cell has been developed by Gratzel and co-workers (O'Regan and Gratzel, 1991; Nazeeruddin et a].,

1993). It is based on a dye-sensitized nanocrystalline, wide band gap semiconductor, usually titanium dioxide. The principal solar cell structure is explained in Fig. 12-33. The photoelectrode consists of a nanoporous film (thickness about 10 pm) of dyecovered TiO, particles, which is deposited on a transparent conducting oxide SnO,: F on glass. The counter electrode also consists of glass with a conducting oxide on which a catalytic amount of platinum is present. In a complete cell, photo- and counterelectrode are clamped together and the space between the electrodes and the porous TiO, nanoparticles is filled with an electrolyte consisting of an organic liquid containing a redox couple, usually iodidehriiodide (I-/Ii). The working principle of the cell is as follows: The light is absorbed by a monolayer of a suitable ruthenium dye, which covers the TiOz surface. Since a dye monolayer on a flat surface absorbs less than 1% of the incoming light, the surface area has to be enlarged by a factor of 1000. This is achieved

766


Figure 12-33. Schematic diagram of a nanocrystalline TiO, dye-sensitized solar cell. Enlarged region shows the working principle and charge carrier transport when the cell is illuminated.

by using TiO, nanoparticles with a diameter of approximately 20 nm. The incoming photons excite electrons i n the dye, which are immediately injected into the TiO, conduction band leaving an oxidized dye molecule. Electrons percolate through the TiO, and are fed into the external circuit. At the counterelectrode triiodide is reduced to iodide by platinum by the uptake of electrons from the external circuit. Iodide is transported through the electrolyte towards the photoelectrode where it reduces the oxidized dye, which is ready again for excitation. Since the basic principle here differs completely from that of semiconductor solar cells, new approaches are required to determine, for instance, the loss mechanisms and how to eliminate them. The most important parameters turned out to be: the mobility and relaxation rate constant of electrons in the TiO,, the electrolyte diffusion constant, the dye content of the cell, and the series resistance of the transparent conductive oxide layers. Efficiencies between 4 and 8% have been reported, but further improvements can certainly be expected (McEvoy and Gratzel, 1994). An advantage of the cell is that it can easily be fabricated and incorporated into a glass structure, for instance, glass windows. Technical difficulties lie in the permanent encapsulation of the electrolyte between the two glass plates.

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General Reading 14th European Photovoltaic Solar Energy Conference: Bedford, U.K.: Stephens & Associates (1997). 25th IEEE Photovoltaic Specialists Conference: New York: IEEE (1996). 1st Int. Con$ on Shaped Crystal Growth, J. Cryst. Growth (1987), 87.

13 New Materials: Gallium Nitride

.

Eicke R Weber and Joachim Kriiger Department of Materials Science and Engineering. University of California. Berkeley. CA. U.S.A. and Materials Science Division. Lawrence Berkeley National Laboratory. Berkeley. CA. U.S.A.

Christian Kisielowski National Center for Electron Microscopy. Lawrence Berkeley National Laboratory. Berkeley. CA. U.S.A.

772 List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 13.1 13.1.1 Applications of GaN and Related Alloys . . . . . . . . . . . . . . . . . 774 13.1.2 Specific Materials Problems of 111-Nitrides . . . . . . . . . . . . . . . . 775 13.2 Growth of GaN and Related Alloys . . . . . . . . . . . . . . . . . . . 777 777 13.2.1 Bulk Growth from Solution . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Hydride Vapor Phase Epitaxy (HVPE) of 111-Nitrides . . . . . . . . . . . 778 13.2.3 Metal-Organic Vapor Phase Epitaxy (MOVPE) of 111-Nitrides . . . . . . 778 13.2.4 Molecular Beam Epitaxy (MBE) of 111-Nitrides . . . . . . . . . . . . . . 780 13.2.5 Epitaxial Lateral Overgrowth (ELOG) of 111-Nitrides . . . . . . . . . . . 781 13.2.6 Laser Lift-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 Defects in 111-Nitrides . . . . . . . . . . . . . . . . . . . . . . . . . . 783 13.3 783 13.3.1 Extended Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 13.3.2 Point Defects and Doping Issues . . . . . . . . . . . . . . . . . . . . . . 13.4 Optical Properties of 111-Nitrides . . . . . . . . . . . . . . . . . . . . 789 789 13.4.1 Bandedge-Related Transitions . . . . . . . . . . . . . . . . . . . . . . . 794 13.4.2 Donor-Acceptor Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 13.4.3 Yellow Luminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 13.4.4 Cubic GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Properties of 111-Nitrides . . . . . . . . . . . . . . . . . . . 797 13.5 13.6 Devices Based on 111-Nitrides . . . . . . . . . . . . . . . . . . . . . . 800 Optical Devices: Light Emitting Diodes (LEDs) and Lasers . . . . . . . 801 13.6.1 13.6.2 Electronic Devices: Field Effect Transistors (FETs) . . . . . . . . . . . . 803 13.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 13.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 13.9

772


List of Symbols and Abbreviations a

b C

d

e E, E (A)

f h

WLO

J k0

m me m (h) n Ndisl

P R T (I

B ’/p h

A CR As0 &

Ad

P Y

r

lattice parameter Burgers vector lattice parameter distance between acceptor states along dislocation line electron charge activation energy acceptor ionization energy occupation fraction of acceptor states longitudinal optical phonon energy total spin momentum Boltzmann constant carrier effective mass free electron mass hole effective mass electron concentration dislocation density vapor pressure distance between donor and acceptor temperature parameter in Varshni equation, Hayne’s rule parameter in Varshni equation, Hayne’s rule exciton-phonon coupling strength axial crystal field splitting spin -orbit splitting dielectric constant Debye screening length electron mobility frequency linewidth

acceptor-bound exciton AX brittle to ductile transition temperature BDT bound exciton BE convergent beam electron diffraction CBED two-dimensional 2D three-dimensional 3D donor-acceptor pair DAP direct current dc DVD-RAM digital video disk random access memory donor with large lattice relaxation, donor-bound exciton DX conduction-band - acceptor eA electron cyclotron resonance ECR


EELS ELOG FE FET FWHM FX HEMT HVPE LD LED LEO LO LT MBE MD-SLS M-I-n MOCVD MOSFET MOVPE MQW ODMR PL RF RT SEM TEM TM VPE w-GaN

electron energy loss spectroscopy epitaxial lateral overgrowth free exciton field effect transistor full width half maximum free exciton high electron mobility transistor hydride vapor phase epitaxy laser diode light emitting diode lateral epitaxial overgrowth longitudinal optical low-temperature molecular beam epitaxy modulation-doped strained layer superlattice metal-insulator-n-semiconductor metal-organic chemical vapor deposition metal-oxide-semiconductor field effect transistor metal-organic vapor phase epitaxy multi quantum well optically detected magnetic resonance photoluminescence radio frequency room temperature scanning electron microscope transmission electron microscope trimethyl vapor phase epitaxy w urtzi te-GaN

773

774


13.1 Introduction The recent surge of interest in GaN and the related group 111-N alloy system (III=gallium, indium, or aluminum) is rapidly establishing this materials system as the third most important semiconductor after silicon and the conventional 111-V semiconductor compounds related to GaAs. This development was triggered by reports on the p-type doping of this wide band gap material by Amano et al. (1989), which was rapidly followed by the fabrication of p/n junctions, light emitting diodes, and laser structures with emission wavelengths ranging from the visible spectrum to the ultraviolet (Nakamura and Fasol, 1997). Since then, any commercially viable 111--nitride technology develops within unprecedented short periods of time. This chapter attempts to provide a concise introduction into this rapidly developing field of research and device development. More detailed discussions can be found in several recent reviews, e.g., Pankove and Moustakas (1998. 1999), Edgar (1994), Edgar et al. ( 1999), Pearton et al. ( 1999), Nakamura and Fasol ( 1997), Orton and Foxon (1998), Monemar (1999), Liu and Lau (1998). and Ambacher (1 998). 13.1.1 Applications of GaN and Related Alloys The unique combination of a direct band gap, which can range from the red for InN to the deep UV for AIN, with bonding strength resistant against recombinationenhanced defect formation and motion, allows efficient light emitting diodes (LEDs) and lasers (LDs) to be realised in the whole visible range (Nakamura and Fasol, 1997). Applications for these solid state emitters range from displays such as brilliant large TV screens on the street or decorative illu-

mination, e.g., in Pachinko gambling machines in Japan, to traffic lights and lasers in high density compact disk drives for optical information storage. For instance, it is expected that the capacity of re-writable DVD-RAM disks will increase from the current 4.7 GB to 15 GB by switching to a blue laser. This gain in information storage capacity will allow 6 h of continuous video storage and is bound to turn over the current market structure for home-video applications. The blue or ultraviolet photons can even stimulate suitable phosphors for the creation of white light, and are expected to soon revolutionize room illumination. The rapid advancements of GaN-based opto-electronic technologies witnessed over the last 10 years can be best demonstrated by a chronological plot of the evolution of the energy conversion efficiency of LEDs in comparison with GaAs-based LEDs (Fig. 13-1). Gallium nitride is also a rare exception in the history of semiconductors with commercial applications way ahead of scientific understanding. Unlike with many other semiconductor systems, GaN-based devices are already available in electronic shops even though comparatively little fundamental insight has been gained into their functioning mechanisms. For example, blue LEDs were already commercialized in 1992, but seven years later the basic nature of the radiative recombination mechanisms in these devices is still the topic of controversial scientific discussions. In addition to optical applications, 111-N heterostructures are very promising candidates for demanding transistor applications. In this field, currently the most intensely developed devices are high power, high frequency microwave amplifiers based on AIGa.N/GaN high electron mobility (HEMT) structures (Nguyen et al., 1998; Sullivan et al., 1998; Wu et al., 1998).

13.1 introduction

AIinGaPIGaP (red, orange, yellow)

n

%B

B u)

101

-

AilnGaPlGaP

f Unfiitered incandescent

775

-

\

e

-E

f Red filtered incandescent

3

Y

8 5 E

1 -

$ P

1965

1970

1975

1980

1985

1990

1995

2000

Time [years] Figure 13-1. Chronological breakdown of the development of the performance of GaN- and GaAs-based LEDs (Ponce and Bour, 1997).

Further applications are currently under consideration in R&D labs, such as visible-transparent (“solar blind”) detectors and radiation-hard electronics. Although it is not yet clear in which of these fields IIInitrides will finally prevail, it is already safe to state that this materials system will take a substantial share of the nonsilicon semiconductor market and thus warrants intense research efforts to improve the materials quality and optimize it for specific applications.

13.1.2 Specific Materials Problems of 111-Nitrides The unusual combination of electronic properties and mechanical stability of 111nitrides has to be paid for by the need to deal with extraordinary materials problems which to date prevent the fabrication of devices with a low density of crystal defects. Commercial

LEDs are known to contain densities of threading dislocations in excess of lo8 cm-2, sometimes even reaching 10” cm-2 (Lester et al., 1995). At the core of this problem is the fact that the high bond strength of GaN results in a melting point near 2800°C. In this temperature range the equilibrium nitrogen partial pressure is extrapolated to be above 45 kbar (4.5GN m-2) (Nakamura and Fasol, 1997; Grzegory and Porowski, 1999), which hitherto prevented any direct experimental determination of the melting point and any single crystal growth from the melt. Therefore device structures can only be obtained by heteroepitaxial deposition on substrates such as sapphire or Sic. Connected with the high melting point is a high brittle-ductile transition temperature so that most thin film deposition has to be performed in the brittle regime, Le., essentially depositing a ceramic material (see Fig. 13-2).

776

13 New Materials:

0.2

I+

0.0

1

Gallium Nitride

MBE (LT-GaAs)

0.2

C 1

1

1

c 3 1

1

1

1

1

1

1

0.0

1

Figure 13-2. Brittle to ductile transition temperature TeDT and temperature of onset of dislocation motion Tplast in selected materials systems. Values are given as fractions of the melting point temperature. Typical growth temperatures of GaN [TMO,--D(Gas)= 1050°C and TMBE,GaN)=7250C] and of GaAs (TGaAs=60O0Cand TLT.GaAr=2000C) are indicated. For further details see text.

The current rapid progress in the development of GaN-based devices is based on experience with thin film growth which is commonly gained in trial and error processes. Consequently, many of the physical processes which control the performance of GaN thin films and heterostructures are “handled” i n some manner, but are not yet fully understood. Stress and strain in GaN thin films is one of these issues. It was recognized early on that large stresses originate from growth on lattice mismatched substrates, sapphire, or S i c , with thermal expansion coefficients that differ from that of GaN (Table 13-1). Film cracking was reported and investigated (Hiramatsu et al., 1993). It could be avoided if the film thickness was kept below 4 pm. Amano et al. (1986) introduced the growth of AlN buffer layers in order to improve the structural quality of the GaN main layers. Dislocation annihilation in the buf-

fer layer region was stimulated by this growth process. Similar results were obtained by growing a GaN buffer layer at temperatures around 700 K (Nakamura, 1991). However, it was not until recently that the

Table 13-1. Comparison of the two most popular substrate materials for hetero-epitaxial growth of GaN. a Material

Lattice parameter a

Lattice mismatch to w-GaN

(.Qb

Thermal expansion coefficient (at 300 K) ( 1 0-6 K-I)

w-GaN AI203 6 H-Sic

3.189 4.758 3.08

5.59 7.5 4.2

- 14%

+4% ~

~~

Note that the GaN hexagon when grown on sapphire is rotated by 30” with respect to the sapphire hexagon. The real lattice mismatch is therefore only - 14%; I A=0.1 nm. a

13.2 Growth of G a N a n d Related Alloys

large impact of buffer layer growth on the stress in the films was discovered (Rieger et al., 1996; Kisielowski et al., 1996b). Moreover, it was pointed out that the growth of buffer layers contributes with a biaxial strain to the overall stress in the thin films which can also be altered by the growth of main layers of different lattice constants (Kisielowski et al., 1996b). Lattice parameters are usually changed by a variation of the native defect concentrations or by the incorporation of dopants and impurities. Such point defects introduce hydrostatic strain components. Thus hydrostatic and biaxial strain components coexist in the films, and are physically of different origin. The biaxial strain comes from the growth on lattice-mismatched substrates with different thermal expansion coefficients. The hydrostatic strain, however, originates from the incorporation of point defects. It was shown that a balance of these strains can be exploited to strain engineer desired film properties (Kisielowski et al., 1996b; Klockenbrink et al., 1997; Fujii et al., 1997; Kisielowski, 1999a). In contrast to all other IIIN semiconductors in practical use, 111-nitridescrystallize preferentially in the hexagonal form with a wurtzite lattice. Therefore the usual thin film growth plane is the densely packed polar (0001) plane which corresponds to the (1 1 1) plane of cubic semiconductors. Thus the polarity of the layers has to be taken into account and antiphase domains with inverted polarity can frequently be observed. Very recently, it was realized that lateral current transport at heteroepitaxial interfaces such as AlGaN/GaN cannot be understood without taking into account the large pyro- and piezoelectric effects caused by the built-in charge polarization at the (0001) interface and its modification by the stress state of the structures (Yu et al., 1997). Besides these problems with structural defects and residual strain, there are very

777

challenging problems with p-type doping, which will be discussed in Sec. 13.3.2, and technological problems such as obtaining good ohmic contacts with p-GaN, which are outside the scope of this chapter. A good summary concerning these problems can be found in recent reviews, e.g., Nakamura and Fasol (1997), Pankove and Moustakas (1998, 1999), Edgar et a]. (1999), and Liu and Lau (1998).

13.2 Growth of GaN and Related Alloys The key factor that has to be considered in the growth of GaN compared with other III/V semiconductors is the high melting point combined with the high nitrogen vapor pressure at elevated temperatures. This property has prevented, up to now, single crystal growth from the melt and results in thin film deposition being a dynamic equilibrium between deposition and decomposition (Newman 1998), as given by the nitrogen desorption rate measured in 1965 by Munir and Searcy (Fig. 13-3). This limits the substrate temperature of all deposition techniques working in a vacuum or with low nitrogen flow rates to lower values compared to deposition techniques working at ambient or enhanced pressure or at high nitrogen flow rates.

13.2.1 Bulk Growth from Solution In the sec. 13.1.2 it was pointed out that the high melting point of GaN combined with the high nitrogen partial pressure at the melting point hitherto prevented any melt growth of single crystal GaN. Nevertheless, the group at the UNIPRESS institute (Warsaw, Poland) succeeded in the solution growth of unseeded GaN single crystal platelets of the order of 1 cm2 in size (Grzegory and Porowski, 1999). Although these


"58

0

08%

I 0

0

Ba, 0

00

0

O 0

I400 1

6.5

I

1

7.5

8.0

1

7.0

1a 1

1200

(OK

to4

I

8.5

( O K - ' )

Figure 13-3. Nitrogen desorption rate measured in 1965byMunirandSearcy.(l atm=1.013x105N m-'.)

represent the best available GaN single crystals, problems with reproducible growth of large numbers of samples have prevented a significant impact of these samples in the field. However, they may serve an important role as reference materials to test the properties of nearly ideal 111-nitrides.

13.2.2 Hydride Vapor Phase Epitaxy (HVPE) of 111-Nitrides The Epitaxial deposition of GaN thin films by halide vapor phase epitaxy (HVPE) has been known for 30 years (Maruska and Tietjen, 1969). The layers obtained by this technique are inferior in structural quality to layers deposited by molecular beam epitaxy (MBE) or metal-organic vapor phase

epitaxy (MOCVD). However, HVPE allows deposition rates reaching 100 pm h-', so that thick layers can be grown as templates for further thin film deposition. This approach might in the future be combined with the recently developed laser lift-off technique (Wong et al., 1998) to obtain large, free-standing films of GaN for homoepitaxial deposition (see Sec. 13.2.6). In conventional HVPE, NH, gas is used as the nitrogen source and gallium is supplied by GaCl gas produced by the reaction of HCl with gallium kept at around 850°C. The lattice strain in HVPE layers deposited on sapphire substrates was found to decrease as a function of layer thickness, with layers of 100 ym thickness being practically strain-free (Hiramatsu et al., 1993). The optical quality of such films is quite remarkable, even early on near-bandedge PL linewidths of 1.5-2.5 meV were reported (Naniwae et al., 1990). For HVPE layers, the absence of the so-called yellow luminescence centered around 2.2-2.3 eV is typical (Molnar et al., 1997; Molnar, 1999). A serious limitation for applications of HVPE-grown GaNlayers is due to the amount of stress incorporated in these layers. The resulting wafer bending currently prevents any consideration of device processing steps that include photolithography. Figure 13-4 shows a direct correlation between the wafer bowing and the energetic position of bandedge-related PL transitions, indicating the varying amount of stress in the HVPE layer (Kruger, 1998). Otherwise, the electrical properties of the film appear to be rather uniform within 30% (Molnar, 1999).

13.2.3 Metal-Organic Vapor Phase Epitaxy (MOVPE) of 111-Nitrides Using metallorganic instead of halide compounds for the group-I11 source in VPE deposition drastically improves the process

13.2 Growth of G a N and Related Alloys

779

Figure 13-4. Relationship between wafer bowing of HVPE-grown GaN at 300 K and stress in the layer as indicated by the shift of photoluminescence lines measured at 4 K .

Position on wafer [mm]

3.420

~

F 3.418 t

1

2 l

-p 3.416 E

0

n

x" Y

c

3.414

= 0 0

a 3.412 3.410

Position on wafer [mm]

control, so that MOVPE (sometimes referred to as MOCVD-metal-organic chemical vapor deposition) is now the preferred thin film deposition technique for advanced III/V devices. This technique was used early on for the deposition of GaN (Manasevit et al., 1971). In particular, the introduction of AlN (Amano et al., 1986) or GaN buffer layers (Nakamura, 1991) lead to dramatic advances in the materials quality and control of n- and p-type doping (Amano et a]., 1989; Nakamura et al., 1991) as well as the possibility to deposit (AI, Ga, In) ternary compounds and heterostructures (It0 et al., 1990; Nakamura and Mukai, 1992; Khan et a]., 1990). It is today the pre-eminent tech-

nique in the fabrication of GaN devices (Nakamura and Fasol, 1997). Detailed reviews of MOVPE growth of 111-nitridescan be found elsewhere (Dupuis, 1997; DenBaars and Keller, 1998). MOVPE deposition is typically carried out at atmospheric pressure, allowing for substrate temperatures in excess of 1000°C for high nitrogen flow rates. On the other hand, the requirement to decompose the constituents at the heated substrate gives a lower limit to the substrate temperature of about 600°C. The detailed growth process is quite complex, involving gas phase reactions of the metallorganic, typically a trimethyl compound such as TMGa, TMAl, or

780


TMIn, and ammonia (NH,), and their decomposition at the heated substrate; a short recent summary has been given by Redwing and Kuech ( 1 999).

13.2.4 Molecular Beam Epitaxy (MBE) of 111-Nitrides Molecular beam epitaxy is the second leading thin film deposition technique for 111-nitrides. However, unlike the case of IIIarsenides or phosphides, there is no direct elemental source of nitrogen available, so that either ammonia (NH,) is thermally decomposed at the substrate surface (Yoshida et al., 1975), or cracking or excitation of nitrogen gas in a plasma source is used. The high corrosivity of ammonia resulted in increasing interest in plasma sources, and several sources using either electron cyclotron resonance (ECR) or radio frequency (RF) generated plasmas are commercially available. However, the possible damage to the growing layer by high-energy nitrogen species makes a third alternative using dc-generated plasmas very attractive (Anders et al., 1996; Newman, 1997).Evidence of damage due to the high-energy nitrogen species

1o6

T

might be the observation of strong deep level photoluminescence in the yellow regime, which was frequently described for GaN grown with ECR or RF plasmas, but is almost completely absent if dc nitrogen sources are used. Using a constriced plasma hollow anode dc source developed at Lawrence Berkeley National Labs, it was possible to grow GaN on sapphire with a width of the excitonic luminescence of only 1.2 meV (Fig. 13-5) practically without any yellow luminescence (Kruger et al., 1997a). A key problem of MBE growth of GaN is the fact that deposition in a vacuum limits the available nitrogen flux rate. Therefore the kinetic competition between deposition and desorption characterized in Fig. 13-3 does not allow the use of high substrate temperatures in MBE growth, a practical upper limit is about 800°C. This limitation results in a reduction of the surface mobility in MBE growth and thus a generally inferior structural quality of MBE grown layers compared with that of MOCVD thin films (Fig. 13-6). It was recently found that the use of bismuth as a surfactant in MBE growth might help considerably to remedy this problem (Fig. 13-7) (Klockenbrink, 1997).

t""""""""""""'"''"l

1o5

3

-mci

Y

io4

&

io3

&

lo2

q

10' 1oo

2.0

2.2

2.4

2.6

2.8

3.0

Energy [eV]

3.2

3.4

3.6

Figure 13-5. 4.2 K photoluminescence spectrum of GaN thin films which were engineered to be stress-free. A line width of 1.2 meV is unusually narrow for hetero-epitaxially grown films only 2 pm thick. [Figure is taken from Kriiger et al. (1997 a).]

13.2 Growth of GaN and Related Alloys 725°C

985°C 1 0 ' ~ .~. ..

, i!

,

,

.

, ,

,

,

, +

,

,

,

,

~

,

,

,

.

I,

,

,

,

(

MBE

105

10.' 1o

-~

781

Figure 13-6. Estimation of a surface diffusion coefficient on GaN (0001). It is argued that gallium ad-atoms limit the surface diffusion which is stress dependent and can be greatly enhanced by using the bismuth surfactant. 3D and 2D13D growth is observed in the indicated temperature ranges. It does not depend on the growth method

10."

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Reciprocal Temperature [1000/K]

On the other hand, MBE growth is further from the thermodynamic equilibrium than MOVPE, so that MBE offers advantages in the deposition of ternary alloys such as InGaN which suffer from phase separation due to a miscibility gap (Singh and Moustakas, 1996). However, the inhomogeneous indium distribution in InGaN active layers of LEDs and LDs is still a topic of great current interest as it might be a beneficial feature allowing carrier localization in regions of high indium content and thus increased recombination probability (Shapiro et al., 2000).

13.2.5 Epitaxial Lateral Overgrowth (ELOG) of 111-Nitrides

Figure 13-7. Comparison of grain sizes between a) an MOCVD-grown GaN (Tgrowth= lO5O0C) and b) an MBE-grown GaN film GaN (Tgrow*=725'C) utilizing the bismuth surfactant technique [Figure is taken from Klockenbrink (1997).]

A novel approach towards the deposition of high-quality layers of GaN on mismatched substrates such as sapphire is the epitaxial lateral overgrowth [ELOG or LEO (lateral epitaxial overgrowth)] (Nam et al., 1997).This technique makes use of the large anisotropy of the G ~ N growth rate in the [OOOl] direction compared with the lateral growth, which is already evident from the platelet shape of solution-grown crystals

782


Figure 13-8. Cross-sectional TEM micrograph of the laterally overgrown GaN layer on an SiO, mask and window area. [Figure is taken from Nakamura ( 1999 a).]

that show a (0001) habit plane (LilientalWeber, 1999). Due to this anisotropy, it is possible to grow high-quality layers on a patterned GaN template, as shown in Figs. 13-8, 9. After growing a first GaN layer, a pattern is formed lithographically, e.g., out of SO,. Further deposition of GaN, e.g., by MOVPE, proceeds initially only in the open windows, but as soon as the GaN thickness exceeds the thickness of the mask layer, horizontal overgrowth becomes dominant. The remaining density of threading dislocations was reported to be below lo6 cm-2 in the overgrown areas between the windows, whereas in the regions above the windows

or along the coalescence lines, higher defect densities can still be found. This approach to high-quality growth of III-nitride heterostructures seems to be a key step towards the realization of longlived, highly efficient laser diodes (Nakamura, 1999a).

13.2.6 Laser Lift-off The quest to obtain large free-standing substrates of GaN has led to another solution of great technological interest: The removal of a heteroepitaxially deposited GaN layer from the substrate by laser lift-off

Figure 13-9. Structure of the Nichia InGaN multi quantum well laser diode grown on an ELOG substrate. [Figure taken from Nakamura et al. (199Q.l

13.3 Defects in Ill-Nitrides

(Kelly et al., 1997; Wong et al., 1998). Illuminating a GaN layer on sapphire from the back side with light of greater photon energy than the GaN bandgap but smaller than that of the sapphire substrate allows the selective deposition of energy just at the layerhubstrate interface. Using sufficiently high laser power, it is possible to bring the GaN at the interface to such a high temperature that it lifts off from the substrate. The most likely mechanism underlying this process is nitrogen evolution from GaN at high temperatures, as exemplified in Fig. 13-3. It has been shown that this process releases the residual stress between layer and substrate while preserving the optical quality of the layer (Fig. 13-10) (Kriiger et al., 1999a). It can be expected that in the future largearea lift-off e.g. of HVPE-grown thick layers makes GaN widely available for homoepitaxial deposition which could have a significant impact on the further development of this technology.

stress free

4

i'""''"'"'""'"'"'~"'""'""'" lifted off

3.450 3.455 3.460 3.465 3.470 3.475 3.480 3.485 3.490

Energy [evl Figure 13-10. PL spectrum (taken at 4 K) of the donor-bound exciton, comparing a MOCVD-grown film on sapphire and a free-standing GaN lifted off from its sapphire substrate. The optical quality of the GaN is not affected by the lift-off procedure, as for instance indicated by the constant line width. The change in energy corresponds to a stress release of 0.4 GPa. [Figure is taken from Kriiger et al. (1999a).]

783

13.3 Defects in 111-Nitrides Despite the astonishing success of commercially available LEDs, the key to further development of the 111-nitride technology lies in the control of the extraordinarily high concentration of defects. A high defect concentration is typical for 111-nitride structures. It originates on the one hand from the high melting point, which relates to the ceramic nature of nitrides (see Fig. 13-2), and on the other hand to the lack of substrates for heteroepitaxial deposition with similar lattice constants and thermal expansion coefficients. In the following, the most important defect problems of 111-nitrides in hexagonal thin film structures will be summarized. A comprehensive analysis of extended defects in solution-grown, bulk GaN platelets has recently been given by Liliental-Weber (1999); details of the defect structure found in cubic GaN deposited on GaAs have been summarized by Kuwano (1999).

13.3.1 Extended Defects Any cross-sectional transmission electron microscopy (TEM) image of thin film GaN which is usually grown on c-plane sapphire or 6H-Sic (0001) reveals an extraordinarily large density of threading dislocations and planar defects, mainly stacking faults and inversion domain boundaries (see Fig. 13.11). Dislocation densities of lo8 cm-2 and more are not uncommon, even in commercially sold LED devices (Lesteret al., 1995). The origin of these defects is the heteroepitaxial growth process, which starts with island nucleation on a highly lattice mismatched substrate, continues with island coalescence during the buffer layer growth, and ends in the propagation of threading defects during deposition of the main layer. It

784


Figure 13-11. [ 1 1201TEM cross section micrograph of a 3 p m thick GaN film. A 20 nrn thin GaN buffer layer was grown at low temperature ( - 800 K ) on top of the sapphire substrate (Kisielowski et al., 1997).

has been argued that the widely varying amount of stress found in the layers after growth is due to different degrees of misfit stress relaxation during growth, combined with the introduction of additional stress during cooling down from the deposition temperature (Kisielowski 1999a). Thus it is highly likely that plasticity during growth and cooling down plays an additional part in the formation of the final defect structure. The character of threading dislocations along the [OOOl] growth direction can be found to be screw-type with b = [OOOl], edge type with b = 1/3 [1120] or mixed with b = 1/3 [1123] (Qian et al., 1995a; Ning et

al., 1996; Wu et al., 1996; Romano et al., 1997; Powell et al., 1993). Whereas the formation of a screw dislocation is not unusual for growth on a hexagonal substrate, the frequent observation of threading dislocations with an edge component might be indicative for the stress relaxation processes discussed above, as only the in-plane component of the Burgers vector contributes to stress relaxation. In addition to these perfect dislocations, partial dislocations have been found at the edge of basal-plane stacking faults (Romano et al., 1997). In addition to dislocations, GaN thin films may contain a high density of planar defects such as stacking faults and inversion domain boundaries. There are three types of basal plane stacking fault possible in the wurtzite geometry (Hirth and Lothe, 1982; Liliental-Weber et al., 1996b). These are usually found close to the interface with the substrate (Romano et al., 1997). Prismatic stacking faults may as well form on { 1210) planes and extend through the layers. Inversion domain boundaries have been found on the (0001) basal plane and on { l O i O ) , { 1 Of 1 }, and { 1 O T 2 ) planes which may extend to the surface of the thin film. In the case of { 10iO) inversion domains, six different variances need to be considered if an inversion of the gallium and nitrogen species as well as translations of different amounts are present (Potin et al., 1997). The energy of formation of basal plane stacking faults has been calculated as only a few meV A-’(Wright, 1997; Stampfl and Van De Walle, 1998). A detailed total energy calculation of the most likely atomic structure of the { 1 OTO}-type inversion domain boundary has recently been given by Northrup et al. (1996). They concluded that only with an additional shift of c/2 [OOOI] of the atom positions across the interface is a low energy inversion domain boundary of this type formed. Convergent beam electron


diffraction (CBED)patterns allow the determination of the polarity of compound semiconductor crystals, as first demonstrated for GaAs (Liliental-Weber and Parechanian-Allen, 1986;Liliental-Weberet al., 1988). A detailed analysis of CBED patterns across { l O i O } inversion domain boundaries of GaN confirmed the inversion of the crystal polarity. Modern electron microscopy techniques such as the exit wave function reconstruction (Thust et al., 1995) allow an independent and direct determination of crystal polarity; for an example of the case of an inversion domain boundary see Fig. 13-12. In addition, this technique gives direct confirmation of the c/2 displacement translation that is present and was proposed by Northrup et al. (1996). Any inversion domain boundary changes the polarity of the crystal. Following the convention established for other IIIN semiconductors, for GaN a (0001) or gallium polarity means surfaces terminated with

785

gallium atoms on top of three bonds inclined to the growth direction, whereas (OOOT) or nitrogen polarity denotes the opposite case with nitrogen atoms on top of three inclined bonds. Whether gallium and nitrogen polar surfaces are indeed gallium and nitrogen terminated, as usually observed for other III/Vs, is still the subject of current studies. The (0001 } surfaces with different polarity have drastically different etching and polishing properties, as first observed with bulk GaN platelets (Liliental-Weber et al., 1996a). The nitrogen polar (OOOi) side is chemically more active and can be chemically polished with e.g., KOH solution, whereas the gallium polar side is quite resistant to chemical attack, so that it is difficult to remove, e.g., polishing damage. A third type of structural defect typical for GaN shows characteristic surface pits (“pinholes”) which are associated with Vshaped structural defects in the thin film. The surface pits can range in size from ten

Figure 13-12. Phase of the electron wave at an (ioio) inversion domain boundary in GaN; [11?0] zone axis. A resolution of about 0.12 nm in the image makes the nitrogen atom appear as a shoulder on the gallium atom. (The inset highlights the atomic positions.) Across the boundary the gallium atoms are displaced by c/8 = 0.06 nm (inversion), and a translation of c/2=0.26 nm is present (Kisielowski 1999b).

786


to several hundred nanometers and are frequently at the termination of “nanopipes” which are typically 5 - 10 nm in diameter, or at the surface termination of inversion domain boundaries. It is still being debated whether all of these nanopipes contain dislocations (Qian et al., 1995 b), or whether a considerable number does not have a displacement vector (Liliental-Weber et al., 1997). The observation of closure of some nanotubes inside the crystal without any threading defect is evidence for the absence of a displacement vector in these cases. The formation of these rather large structural defects might be triggered by several mechanisms. Liliental-Weber et al. (1997) point out that the accumulation of impurities such as oxygen or dopant elements can start the formation of nanopipes or pinholes. In strained layer systems, there is evidence that strain accumulation might play a role in the formation process (Kisielowski, 1999a). A key factor i n the development of faceted pinholes seems to be the different crystal growth rates on planes of different crystallographic orientation, with the slowest growth rate found for { 1 OT 1 ) polar planes (Liliental-Weber et al., 1997). Other forms of extended defects which are especially important in 111-nitride based heterostructure devices are strain inhomogeneities, interface roughness, and composition fluctuation of ternary alloy layers such as AlGaN or InGaN. New instrumental possibilities of modem high-resolution electron microscopes open opportunities in the analysis of these types of imperfection (see Fig. 13-13). Quantitative analysis of the interatomic spacing and thus the local strain in combination with a chemical analysis by electron energy loss spectroscopy (EELS) and Rutherford backscattering allows calibration of the strain i n terms of composition on an atomic scale, i.e., with a lateral resolution of less then 1 nm. Such an

analysis is based on cross-sectional TEM samples, Le., it determines averages of a column of up to several hundred atoms in a stack parallel to the electron beam. It is now for the first time possible to profile and map atom displacements as small as 1 pm also in noncubic systems (Kisielowski et al., 1998).

Figure 13-13. a ) High resolution lattice image of an AIGaN/GaN barrier structure; [11?0] zone axis, b) EELS and strain profiles of the area depicted in a). The c lattice parameter is profiled and is a line scan along the c-direction across the barrier layer. c) Strain mapping (c-lattice parameter) over the area of the lattice image depicted in a). [Data taken from Kisielowski ( 1999 b).]


13.3.2 Point Defects and Doping Issues This section will focus on the point defects issues that are relevant for p- and ntype doping of GaN; further discussions of point defects which are studied in optical and electrical measurements are included in Sec. 13.4 and 13.5. Before discussing p- and n-type doping of GaN, a general remark concerning the distinction between hydrogenic and localized energy levels seems to be necessary. Such a distinction should be based only on the nature of the states involved, but it is often confused with the distinction between shallow and deep levels, which is all too often based simply on the magnitude of the ionization energy. Any charge in a semiconductor is expected to form bound states with carriers that are hydrogenic in nature, Le., with a small activation energy and a delocalized wave function. Scanning tunneling microscopy can directly image the attraction of carriers to charged impurity atoms [see, e.g., Zheng et al. 1994)], and the ionization energy depends mainly on the effective mass and the dielectric constant of the material, modified by a small central cell correction. In addition, lattice defects and impurities can form localized states whose ionization energy might place the corresponding level deep in the energy gap, close to the conduction or valence band, or energetically degenerate in one of the bands. Well-known examples for the latter case are the lattice relaxed DX-state of silicon donors in GaAs or the PI, antisite defect in InP (Dreszer et al., 1993). Therefore all discussions of energy levels of shallow dopants in GaN, and especially the discussion of state-of-the-art first principle calculations, have to distinguish whether they are dealing with localized states or hydrogenic states, a distinction that has sometimes been neglected.

787

A viable GaN technology could not be developed in the 1970s and 1980s mainly due to the lack of p-conducting material. During these early years, unintentional n-type doping by impurities or native defects, and autocompensation of acceptors in the wide gap material, were speculated as the origin of this problem. It is worth noting that Pankove, one of the early pioneers of GaN development (Pankove and Berkeyheiser, 1974), was also the first to study acceptor passivation by hydrogen in another semiconductor, silicon (Pankove et al., 1985). However, only the sustained efforts of Akasaki and co-workers finally achieved p-type conductivity in GaN. This happened accidentally by applying low energy electron beam irradiation in an SEM to magnesium-doped MOCVD grown GaN (Amano et al., 1989) which removed what is today believed to have been the main obstacle, Le., hydrogen passivation of the acceptors (Van Vechten et al., 1992; Neugebauer and Van der Walle, 1995; Gotz et al., 1996c, d). Later it was shown by Nakamura et al. (1992 a) that a simple post-growth heat treatment of MOCVD grown p-doped layers activates the magnesium acceptors, most probably by thermally breaking the acceptor-hydrogen bond, as had been demonstrated much earlier by Pankove for boron acceptors in silicon (Pankove et al., 1985). Today, p-type doping by magnesium can be routinely achieved; an upper limit seems to be around 10l8 ~ m - and ~ , due to the absence of hydrogen in MBE growth no postgrowth heat treatment is necessary (Moustakas and Molnar, 1993; Brandt et al., 1994). However, the large acceptor ionization energy measured electrically to be about E ( A ) = 160-180 meV (Gotz et al., 1996b; Kim et al., 1996) requires the incorporation of about lo2' cm-3 magnesium atoms on gallium sites in order to achieve the highest hole concentrations, and it is not clear

788


whether a higher concentration of electrically active, thermally stable, and uncompensated magnesium acceptors may ever be achieved. There is not yet an universally accepted set of values for the anisotropic hole effective mass (Suzuki and Uenoyama, 1999). The values under current discussion of about m (h) = (0.8 f 0.2) m e would result in E ( A )= 120 & 40 meV, but using a high frequency dielectric constant rather than the static value could easily shift the hydrogenic acceptor binding energy into the range experimentally observed for magnesium doped GaN (Orton and Foxon, 1999). Thus this acceptor level is most probably the true hydrogenic level, rather than a deep localized level which could be substantially different for other shallow acceptor impurities. It should be noted that many theoretical calculations of energy levels using typically the local density approximation do not attempt to calculate the hydrogenic level, which would require an accurate calculation of the valence band structure, but are rather aimed at localized levels which might turn out to be more shallow for a specific impurity, but would not be relevant if the hydrogenic state is the deeper ground state. In view of this discussion, it is not surprising that all attempts to use alternative acceptor dopants such as zinc, beryllium, or calcium failed to produce p-GaN with higher hole concentrations (Orton and Foxon, 1999). Yamamoto and Katayama-Yoshida ( 1997, 1999) suggested that donor-acceptor codoping might be a promising strategy, essentially by increasing the solid solubility limit through co-doping. An early result of a metallic-like p-type phase, obtained in cubic GaN containing high concentrations of beryllium and oxygen (Brandt et al., 1996), seems to confirm this mechanism, but convincing, more direct experimental results

demonstrating new record hole concentrations have yet to be presented. For several decades, undoped GaN grown mostly by vapor phase epitaxy was n-type conducting, due to a hitherto unknown background dopant. Due to the high vapor pressure of nitrogen (see Sect. 13.2), the crystals were generally assumed to be nitrogen deficient, and nitrogen vacancies were speculated to be the source of n-type conductivity. Based on a simple ionic model, groupI11 vacancies in a IIIN compound are expected to be acceptors, and V vacancies to be donors. State-of-the-art local density calculations support this notion, but predict formation energies in n-GaN that are too high to account for the observed electron concentrations (Neugebauer and Van der Walle, 1994,1996; Boguslawski et al., 1995). A second widely discussed source of unintentional n-type conductivity is donor impurities, especially the group VI impurity oxygen (Seifert et al., 1983; Chung and Gershenzon, 1992; Boguslawski et al., 1995). Today, it is well known that oxygen forms a shallow donor level that in GaN may even be degenerate with the conduction band (Wetzel et al., 1997; Van de Walle, 1998). Moreover, chemical analysis of typical undoped GaN crystals revealed oxygen concentration levels comparable or above the ntype carrier concentrations, so that this doping mechanism is now widely accepted. However, there might be specific cases in which unintentional silicon doping can add to the n-type conductivity of “undoped’ GaN (Gotz et al., 1996b). A direct distinction between oxygen and silicon shallow donors can be made with the help of hydrostatic pressure experiments: oxygen forms a localized state in GaN, resulting from a large outward relaxation of the small oxygen atom in a gallium vacancy. This localized state is in the conduction band, but enters the band gap upon the ap-

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789

13.4 Optical Properties of Ill-Nitrides

plication of hydrostatic pressure or alloying with aluminum in AIGaN. Then it forms a deep donor level (Wetzel et al., 1997; Van de Walle, 1998), similar to the DX-behavior of silicon and other n-type dopants in GaAs, AlGaAs and other compound semiconductors [see, e.g., Malloy and Khachaturyan 1993)l. Silicon donors do not seem to show this transition at the pressures or alloy compositions studied thus far (Wetzel et al., 1997; Van de Walle, 1998). With the shallow hydrogenic silicon donor level found to be 12-17 meV below the GaN conduction band edge (Gotz et al., 1996b), electron concentrations up to or even exceeding 1O2' cm-3 can be achieved. A serious problem for the highest electron concentrations is caused by the lattice strain due to the silicon dopant atoms, which can result in cracking of the doped film (Ager et al., 1997). This puts a limitation on either the doping level or the thickness of the n+ layer.

13.4 Optical Properties of 111-Nitrides The optical properties of GaN have been mostly studied by photoluminescence, photoreflectivity, and Raman spectroscopy. The following sections focus on the various aspects of photoluminescence (PL) of thin hetero-epitaxially grown GaN films. A general overview on photoluminescence of semiconductors is given by Gilliland (1997). Excellent reviews on GaN specific topics can be found in Monemar ( 1999) and Orton and Foxon (1998).

13.4.1 Bandedge-Related Transitions Due to several peculiarities of the GaN system, the low temperature near band-gap region of the hexagonal GaN phase is rath-

er complex in comparison to other cubic semiconductor systems, for instance, GaAs. First, caused by the hexagonal crystal symmetry, the valence bands near k = 0 are split into three different bands, labeled A, B, and C (Fig. 13-14). The energetic splitting between these bands is given by the spin-orbit splitting A,, and the axial crystal field splitting ACR. Though in theory only A,, should depend on the built-in strain, it has been found that both splittings A,, and ACR vary as a rather complicated function of strain (Gil et al., 1995). For a homo-epitaxially grown GaN film which is believed to be unstrained, these parameters are found to be As,= 19.7k1.5 meV and A,,=9.3* 0.3 meV (Koronaet a]., 1996). Consequently, the free exciton is split into three transitions labeled FXA, FX, , and FX,. Depending on the amount of built-in strain, the spacings between the respective free exciton transitions are FXA-FXB = - 15 to 9 meV andFXA-FX,=8 to50meV(Fig. 13-15.). Principally, these values could be utilized to determine the built-in strain, but in practice FXB and FX,, in particular, are not well enough resolved to allow for an accurate measurement by photoluminescence. Biaxial stress is known to shift bandedgerelated transitions. Depending on the

Wurhite

Zinc Blende

Wurtzite

r,

- -

r, -

-

r,

Simple Group

r.

Double Group

Figure 13-14. Sketch of the symmetries of the conduction and valence band of the zincblende and the wurtzite direct band-gap semiconductor. (r,:crystal field symmetry) [Plot is taken from Gil (1999).]

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790


'4

dilatation compression 3540

0 SIC Substrate

3440

3460

3480

3500

Energy of A line (mev) Figure 13-15. Energies of the B- and C-exciton transitions as a function of the position of the A-exciton. [Plot is taken from Gil (1999).]

growth method, growth conditions, the substrate material, and the thickness of the GaN epilayer, bandedge-related PL lines may shift by more than 40 meV. Experimentally determined calibration factors range from 23 meV GPa-] (Perry et al., 1997) to 27 meV GPa-' (Kisielowski et al., 1997; Rieger et al., 1996). As theoretically predicted (Kisielowski et al. 1997) and experimentally verified (Kruger et al., 1998), the largest contribution to the biaxial stress components of hetero-epitaxially grown GaN stems from the mismatch of thermal expansion between substrate and GaN epilayer and the mismatch in lattice constants between both materials. Additionally, a smaller stress contribution has been shown to arise from (intrinsic) point defects (Kriiger et al., 1998). In a bulk crystal, these defects should introduce mere changes of the lattice constant. Here, the presence of the constraining substrate translates this effect into a biaxial

stress component. At helium temperatures, most authors find the free exciton position to be FXA= 3.475 k0.003 eV for a strainfree crystal. The only significant deviation from this value has been reported by Skromme et al. (1997) who by measuring the curvature of GaN samples grown on sapphire and S i c found the strain-free position for FXA to be 3.468 f 0.002 eV. The low temperature PL spectrum of GaN is dominated by the free exciton transition, if the GaN crystal is of superior optical quality (Shan et al., 1995). In most cases, however, the free exciton is bound to a neutral donor in n-type crystals. The donor-bound exciton transition DoX is red-shifted by 6 +: 0.5 meV with respect to the free exciton transition FXA (Fig. 13-16). The free and bound excitons are usually accompanied by phonon replicas, which depending on the optical quality of the crystal can be observed up to the 6th order (Kovalev et al., 1996). Owing to the strong ionic character of GaN, the longitudinal optical phonon energy h vLo=91.5 meV is rather large in comparison to that for other 111-V semiconductor systems (GaAs: h vLo= 36 meV). The chemical nature of the dominant donor in GaN has been controversially discussed. Independent of the growth method, undoped GaN is usually n-type. Initially, loss of nitrogen during growth was held responsible for the formation of nitrogen vacancies V,, which are believed to be donor defects. On the other hand, theoretical calculations by Neugebauer and Van de Walle (1994) and Boguslawski et al. (1995) have shown that the formation of V, is less favorable in n-type GaN. The current understanding in the literature assumes unintentional contamination with oxygen to be the major source of free n-type carriers. Oxyen, most likely on the substitutional nitrogen lattice site (ON),introduces a shallow level with an optical activation energy of 35 meV


5oooo

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exciton-

D BE 3.4718 eV

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exciton-

-Free

n Y

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D BE 3.4709 eV

4oooo-

1

e

(d

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3oooo

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GaN on GaN substrate MBE, undoped 0.3mm layer PL temperature 4.2K excitation HeCd 1 mW A BE 3.4663 eV

CI

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I 1 ' I . I

.

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*

'

.

I

'

I

*

*

*

I

*

'

.

)

*

3.445 3.450 3.455 3.460 3.465 3.470 3.475 3.480 3.485 3.490 3.495 3. 00

ENERGY [eV] Figure 13-16.Luminescing near band-gap transitions in hetero-epitaxially n-type GaN. ABE: Acceptor bound exciton, DBE: donor bound exciton (D'X), FEA: free exciton A (FX,), FEB: free exciton B (FX,). [Spectrum is taken from Monemar (1998).]

(Meyer et al., 1995). The best resolved analysis of the near band-gap PL spectrum has been conducted by Skromme et al. (1999). The authors find as many as five different donor bound exciton transitions, but are not in the position to assign all of them to a particular defect or impurity. Besides the dependence on stress, the energetic position of the exciton lines moves with temperature, as they follow the energetic band gap. The donor-bound exciton is thermally delocalized with increasing temperature and disappears around 150 K. Due to the rather high exciton binding energy of EFX=25 k 1 meV, the free exciton is still visible at room temperature (Fig. 13-17). Typical line widths are about 35 meV (FWHM), which can make line assignment rather difficult at elevated temperatures. Herr et al. (1998), however, were even able to follow the FX transition up to 1200 K without any indication of contributions from band-to-band transitions. The line width rapidly increases with temperature,

3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 Energy

[evl

Figure 13-17. Temperature dependence of the near band-gap luminescence transitions. Visible are the donor-bound exciton DX and the two free excitons FX (A) and FX (B). [Spectrum is taken from Kruger (1998).]

792

13

New Materials: Gallium Nitride

which is again a consequence of the strong ionic character of the Ga-N bond. Generally, the temperature dependence of the line width can be broken down into three major regimes (Fig. 13-18). At cryogenic temperis dominated by atures, the line width interface roughness, the exciton-exciton interaction, exciton scattering by impurities (Viswanath et al., 1998), and the amount of strain present in the material. The intermediate region is given by the coupling strength Yph of the interaction between excitons and acoustical phonons. For temperatures above 240 K, the line width is entirely dominated by the exciton interaction with longitudinal optical phonons. Interaction parameters measured for various samples are depicted i n Table 13-2.

r,

r,,

Numerous publications have attempted to determine the temperature dependence of

0

100

200

300 400 500

600

Temperature [K] Figure 13-18. Temperature dependence of the free A-exciton line width ( h i i g e r et al,, 1 9 9 9 ~ )For . explanation of symbols, see text.

Table 13-2. Comparison of interaction parameters determining the temperature dependence of the PL line width in GaN. Experimental technique Photoluminescence of FX ( A ) Photoluminescence of FX (A) Photoreflectivity FX (A) and FX (B) Photoreflectivity FX ( A ) and FX (B)

ypha

rLobReference

( w V ) (mev)

31

500

14

395

30

416

30

400

Kruger et ai. (1999a) Viswanath et al. (1998) Korona et al. (1999) Chichibu et al. (1999)

a Coupling strength of excitons with acoustic phonons; coupling strength of excitons with LO pho-

nons.

the GaN band gap between cryogenic and room temperature by various optical methods, including photoluminescence and reflectivity. The huge differences i n the observed dependencies have been ascribed to various material and growth parameters, such as the growth method, post-growth cooling down rate, sample thickness, to name a few. None of these publications, however, is in the position to sort out one particular parameter and to describe or even explain its influence on the GaN band gap temperature dependence. A few experimental results are presented in Fig. 13-19a. It is surprising to notice that even fundamental properties, such as the influence of the mismatch of the thermal expansion coefficients between GaN film and the various substrate materials on the GaN band gap temperature dependence, have not yet been conclusively explained. For instance, Fig. 13-19a shows the temperature dependence of the GaN band gap for films grown on S i c , bulk GaN, and sapphire. Though the variation with temperature can be qualitatively explained by the respective differences in the thermal expansion coefficient of GaN and Sichap-


793

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3.50

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U

p" z

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3.46

3.44

3.42 0

40

80 120 160 200 240 280 320 Temperature [K]

0

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80 120 160 200 240 280 320 Temperature (K)

Figure 13-19. Temperature dependence of the GaN band-gap between 4 K and 300 K. Data were determined by photoluminescence following the free A-exciton position and adding the free exciton optical binding energy of 25 meV. a) Comparison of GaN thin films grown on either Sic, bulk-grown GaN, or sapphire. [Data taken from Buyanowa et al. (1996).] b) Results of GaN films assumed to be stress-free. Monemar (1974) depicts a thick HVPE-grown GaN film, Korona et al. (1996) have measured an MOCVD grown film on bulk GaN, and Kriiger et al. (1999a) present results of a free-standing MOCVD-grown GaN layer lifted off from its sapphire substrate.

phire, the observed absolute differences are too large. Further, the crossover at about 240 K cannot be explained at all. For this reason, authors have followed several strategies to eliminate the conceivable influence of the substrate: Monemar (1974) has measured a 100pm thick HVPE-grown film which is known to be stress-free, Korona et al. (1996) have deposited a thin GaN film by MOVPE on a bulk GaN substrate, and Kriiger et al. (1999 a) have measured a freestanding 2 pm thick MOCVD grown film which has been lifted off from its sapphire substrate by a laser-assisted technique (Fig. 13-19b). Apparently, the differences in the temperature dependence obtained by the several authors are much less than in the first case.

Generally, the temperature dependence of the GaN band gap has been described by the semi-empirically found Varshni formula (Varshni, 1967)

a T~ E ( T )= E ( T = 0) - B'T

As revealed in Table 13-3, the quality and growth history of the investigated GaN crystals is still too variable as to allow for a conclusive determination of these parameters. The typical line width of excitonic lines of MBE-, MOCVD-, or HVPE-grown films at helium temperatures is typically between 2 and 6 meV. This value is too large to correctly identify optical transitions related to the band gap. It has to be assumed that stress is the main broadening mechanism for band-

794


Table 13-3. Comparison of Varshni parameters determined for various GaN materials. Experimental technique

a(104eVK-')

B(K)

Reference

5 .O 52 5.08 9.39 8.32 10.9 5.66

400 450 996 772 835.6 1 I94 737.9 1187.4 1570

Viswanath et al. (1998) Viswanath et al. (1998) Monemar (1974) Monemar (1974) Shan et al. (1995) Shan et al. (1995) Manasreh (1996) Manasreh ( 1 996) Kriiger et al. (1999 a)

Photoluminescence of FX (A) FX (B) PL excitation (100 pm thick film) Photoabsorption Photoreflection of FX (A) FX ( B ) Absorption of MOCVD grown GaN Absorption of MBE grown GaN PL of FX (A) MOCVD, lifted off from sapphire

edge related PL transitions. For hetero-epitaxially grown films on sapphire, the smallest line width for excitonic lines in GaN has been reported as 1.2 meV (FWHM) and has been achieved for an only 2 pm thick film which is under stress-free conditions at helium temperatures (Fig. 13-5, Kruger et al., 1997 a). It appears that line widths can only be further reduced by homo-epitaxial growth on bulk GaN. A very recent effort reports on a record line width of only FWHM= 100 peV (!) for a 1.5 ym thick film grown by MOVPE on bulk GaN (Kornitzer et al., 1999; Kirchner et al., 1999). Thus a detailed analysis and line assignment of bandedgerelated transitions can be expected very soon. The excitonic lines related to acceptorbound states are less profoundly researched. Merz et al. (1996) report on two lines being 34 +4 meV and 19 + 4 meV below the free excitonic FX, transition which they attribute to the zinc and magnesium acceptor, respectively. This assignment is based upon the Haynes' rule (Haynes 1960), which suggests that the localization energy of an acceptor-bound exciton EL, should be one tenth of the acceptor binding energy

ELD= E F X A- EDx= a ED EL,

-EAX =

PEA

with a = 0.2 for donor and P= 0.1 for accep-

tor-bond excitons.

11.56 13.56

13.4.2 Donor- Acceptor Pairs

Another group of signals, visible in most GaN crystals, is the donor-acceptor pair (DAP) recombination (Fig. 13-20); an electron trapped at a donor is recombining with a hole trapped at an acceptor. The actual recombination energy is then given by

The last term accounts for the Coulomb interaction, because donor and acceptor are oppositely charged after the recombination, R is the distance between the donor and acceptor involved. Since this transition sums up all possible distances between donors and acceptors, the signal tends to be much broader than, for instance, the excitonic luminescence (Fig. 13-20). The most dominant DAP signal in slightly n-type GaN can usually be detected around 3.27 eV (Dingle and Ilegims, 1971). It is usually accompanied by two or more phonon replicas (hvLO=91.5meV). The natures of the involved donor and acceptor are still under debate. For the acceptor, it has been observed that the DAP signal correlates with the concentration of magnesium doping (Ilegims and Dingle, 1973), but it has also been argued from early on that this spectrum might arise from a residual acceptor (Dingle and

13.4 Optical Properties of Ill-Nitrides I

~

"

'

I

'

"

'

I

'

'

~

~

I

"

"

I

'

Donor-Acceptor Pair transition (DAP) I

I

Donor bound exciton transition (DX)

.

795

~ Figure 13-20. Donor-acceptor pair transition in undoped n-type GaN measured at 4 K. Visible are the DAP and the eA transition (Kriiger, 1996).

Band-Acceptor transition (eA) :

3.0

"

.

~

~

3.1

'

"

"

'

'

'

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3.2 3.3 Energy [evl

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3.4

Ilegims, 197 l), probably carbon or a native defect associated with the incorporation of magnesium. Fischer et a]. (1995) found the ionization energies of carbon and magnesium to differ by 20 meV at the most. Benefiting from the high quality of homo-epitaxially grown GaN films, Baranowski and Porowski (1996) were able to resolve discrete transitions for neighboring donor-acceptor pairs. The authors' analysis of the energetic distribution of these PL signals led to the conclusion that the involved donor is a defect on the nitrogen lattice site, most likely being the nitrogen vacancy. When either the free n-type carrier density exceeds [n] = 1 x 10l8cm-3 or the temperature is raised, the comparatively shallow donor becomes ionized and the signal shifts to higher energies (Fig. 13-20). Generally taken as an indication of a donor-acceptor pair, the transition shifts with temperature following this formula

3.5

13.4.3 Yellow Luminescence The yellow luminescence (YL) is a broad band centered around 2.25 eV, which is often visible to the nakedeye. This line is present in almost any GaN material and its intensity is regarded as a measure for the optical materials quality, since excellent materials don't show any YL. Though this "parasitic" transition has been shown not to affect the efficiency of high carrier injection operated opto-electronic devices (Grieshaber et al., 1996) its chemical nature and its exact transition model have been subject to a long-standing debate. Two competing models for the luminescing transitions have been controversially debated: Based upon ODMR (optically detected magnetic resonance) experiments, Glaser et al. (1995) proposed a transition between a deep double donor (1 eV below the conduction band) and an effective-mass acceptor. The other model by Ogino and Aoki (1980), now being widely accepted, assumes the transition from a shallow donor to a deep acceptor level (Kaufmann et a]., 1999).The chemical origin of the deep level has been equally debated. Among the suggested defects are a

796


complex between a gallium vacancy and a carbon atom (Ogino and Aoki, 1980), an NG, antisite defect (Suski et al., 1995), and a complex between a gallium vacancy and oxygen (Neugebauer and Van der Walle, 1996). Recently, theoretical as well as experimental studies have evidenced the gallium vacancy V,, as the most likely candidate for the deep acceptor. First principle calculations by Neugebauer and Van de Walle (1996) have shown that the gallium vacancy VGAis expected to be a deep acceptor and that, consequently, its formation energy should decrease with increasing Fermi level. This goes hand in hand with the experimental fact that the YL has so far only been observed in ntype GaN materials. Recently, the correlation between the YL and the gallium vacancy defect has also been proven experimentally. Positron annihilation spectroscopy being sensitive to negatively charged vacancy defects has shown a strong correlation between the density of gallium vacancies and the intensity of the YL (Saarinen et a]., 1997). 13.4.4 Cubic GaN Though the cubic phase is generally less well characterized than the wurtzite one. all

major optical transitions in the cubic phase have already been identified by photo- and cathodoluminescence. The analyzed material has mostly been grown on cubic substrate materials, such as MgO (RamirezFlorez et a]., 1994), p - S i c (Okumura et a]., 1994),(100)Si(Leietal., 1991),(lOO)GaP (Cheng et al., 1995; Orton, 1995), or on (001) GaAs (Strite et al., 1991; As et al., 1997; Menniger et al., 1996a, b). In most cases, the layers were deposited by MBE, but it is reasonable to assume that it is the lower growth temperature rather than the growth method itself which favors nucleation of the cubic phase. When deposited by MBE on sapphire, the predominant hexagonal phase may exhibit cubic inclusions whose optical transitions have been studied by Strauss et al. (1997), Andrianov et al. ( 1996) and Kriiger et al. ( 1996b). A typical PL spectrum is shown in Fig. 13-2 1. All the references observed the boundexciton luminescence at DoX = 3.268k 0.006 eV. Further, a strong donor-acceptor pairtransitionisreportedat3.165k0.015 eV. The chemical nature of the involved acceptor is not known, but its energetic depth is only 130 meV. This is considerably lower than the lowest known acceptor level in the

DX hexa. cubic

cubic

'250

2.8

2.9

3.0

3.1

'50

3.2

Energy [evl

3.3

3.4

3.5

Figure 13-21. PL (at 4 K) of a predominantly hexagonal GaN film grown by MBE on sapphire with cubic inclusions. Besides the near bandgap spectrum of the hexagonal phase, the donor-bound exciton and the DAP of the cubic phase are also recognizable. [Data taken from Kruger et al. ( 1997 b).]

13.5 Electrical Properties of Ill-Nitrides

wurtzite phase and, though being speculative at the present time, it presents one of the most interesting features of the cubic GaN phase. There is general agreement that the band gap of cubic GaN is 203 meV smaller than that of the wurtzite phase. The generally accepted values at room temperature are:

Eg (cubic)=3.21k0.02 eV (Ramirez-Florez et al., 1994)

E, (hexag.) = 3.41 k 0.02 eV (Monemar, 1974).

13.5 Electrical Properties of III-Nitrides The electrical quality of a semiconducting material is usually indicated by the majority carrier (low field) mobility, which is commonly measured by the Hall effect. Thus this section focuses on carrier mobility issues in GaN. It should be explicitly noted that the transversal mobility is analyzed, though most LED and laser designs are operated in vertical carrier transport. Early studies of the transport properties of GaN have severely suffered from the low quality of the material available at the time; a summary of these results can be found in the review by Gaskill et al. (1994). Additional complications initially arose from the fact that hetero-epitaxially grown GaN has an inhomogeneous depth-profile: the defect-rich interface region may supply an extra conduction path, which may change the apparent carrier concentration as well as the mobility by several orders of magnitude. This phenomenon has been found to be most pronounced at cryogenic temperatures. Earlier studies did not take this effect into account; reliable data can be found in recent publications (Dhar and Ghosh, 1999; Look and Molnar, 1997).

797

The highest electron mobility at room temperature ever reported for a hetero-epitaxially grown GaN film has been given by Molnar et al. (1995) for an HVPE-grown film, p= 950 cm2 V-' s-l (at 300 K). Electron Hall mobilities in MBE-grown material are genuinely smaller. This is most probably due to the rather low growth temperature, which results in hexagonal columns of typically 1 pm in diameter. Grain boundaries are believed to be effective carrier scattering centers. The highest reported mobility for a plasma-assisted MBE-grown GaN layer deposited between 700 and 800°C is p=300 cm2 V-' s-l(300 K)(Ngetal., 1998). Ammonia-MBE, which allows growth at a higher temperature (860-920°C) is shown to result in mobilities in excess of 550 cm' V-' s-' (Tang and Webb, 1999). Bulk-grown GaN is usually strongly n-type and reveals a typical mobility of p= 100 cm2 v-' s-l with a background n-type concentration of [n]= 1019 cm-3 (Porowski, 1996). Because of the carrier degeneracy, the carrier concentration is temperature-independent. Principally, the carrier mobility at room temperature is limited by the amount of ionized donor and acceptor states and the compensation ratio. Theoretically (Dhar and Ghosh, 1999), a mobility value of about 1350 cm2 V-' s-l should be achievable for carrier concentrations below 10l6 ~ m - In ~. practice, however, concentrations around lo" cm-3 are typically achieved, which should yield a maximum mobility of around 780 cm2 V-' s-*. The temperature dependence of the ntype carrier mobility can be broken down in the following scattering mechanisms (Dhar and Ghosh, 1999). Since GaN is a strong polar semiconductor, the electron-phonon scattering dominates at most temperatures. Within this regime, optical phonon scattering contributes the most, since it is the only nonelastic process. The scattering rate is

798

13 N e w Materials: Gallium Nitride

one order of magnitude larger than that of GaAs, being caused by the higher ionicity of GaN. The electron-acoustic phonon and the piezoelectric scattering have been found to be negligible. In the regime of optical phonon scattering, the mobility varies with T-3‘2,In the low temperature case, impurity scattering presents the dominant process and follows a T3’* dependence. For further details, see Dhar and Ghosh (1999) and references therein. Experimentally, the temperature dependence of the carrier concentration and mobility is already well studied; for recent reviews see Look (1 999), Gotz and Johnson (1999). The n-type carrier concentration of silicon-doped n-type material is thermally activated and two activation energies have been found (Fig. 13-22). One, ranging from 12 to 16 meV, is attributed to the silicon donor. The second level (32-37 meV) occurs in samples with a typical density of about 3 -6 x 1 O I 9 cm-? and is speculated to be associated with the unintentional oxygen donor. After heat treatment to activate the magnesium dopant, the temperature dependence

(a)

Temperature (K)

500

200 150

of the p-type carrier can be simulated under the assumption of a single acceptor; the typical thermal activation energy is between 160 and 175 meV (Fig. 13-23). The actual acceptor activation energies depend on the incorporated magnesium concentration. As experimentally confirmed by many publications and theoretically explained by Dhar and Ghosh (1999), the electron mobility increases with decreasing amounts of n-type dopants. It was not until very recently, however, that the influence of (charged) dislocations on electron scattering was experimentally verified (Ng et al., 1998) and theoretically calculated [Weinmann et al., ( 1998), Fig. 13-24]. It has been argued that dislocations with an edge component introduce acceptor states (as is well known from the case of silicon). For a GaN film with a dislocation density of 10’’ cm-*, the number of dangling bonds has been estimated to be 2x10” ~ m - In ~ .n-type crystals, by attracting electrons from the surrounding area, the dislocation line becomes negatively charged. Thus a space charge region is formed around the dislocation line. which

100 I

MOCVDgrowr GaN

lo’’ 2

4

6

8

10001T (1IK)

sample#:

10

12

100

o

1

A

2

300

Temperature (K)

600

Figure 13-22. a) Electron concentration vs. reciprocal temperature and b) Hall mobility vs. temperature for an undoped (sample #1) and silicon-doped samples (#2 -4). [Data taken from Gotz and Johnson (1999).]

13.5 Electrical Properties of Ill-Nitrides

(a) l o"

500

Temperature (K) 300 200 150

Figure 13-23. a) Hole concentration vs. reciprocal temperature and b) Hall mobility vs. temperature for magnesium-doped, p-type GaN samples. [Data taken from Gotz and Johnson (1999).]

(b)

1O'O

5 loi' 0

Y

-

.-5

sP

r"

799

1017

IOi5 8

I 0" 0

10''

I

2

3

4

5

6

7

IOOOlT (IlK)

.

, . . ,

300 600 Temperature (K)

Figure 13-24. Electron mobility vs. net carrier concentration of n-type GaN samples of varying dislocation density. The curves in the low carrier concentration regions are theoretical curves fitted to Eqs. (13-1) and (13-2). [Data taken from Ng et al. (1998).]

scatters electrons thereby reducing the lateral electron mobility. It depends on the balance between (dopant) donor concentration and dislocation density as to whether this effect is visible or not. As shown in Fig. 13-25, in the high temperature range, the mobility will approach the T-3'2 dependence due to LO phonon scattering. For a high concentration of free carriers, the low temperature regime is governed by pure ionized impur-

ity scattering following a T3I2dependence (Fig. 13-25). In the opposite scenario, the low temperature regime is determined by dislocation scattering. Both the carrier concentration and mobility are thermally activated, the latter one following the formula given by Podor (1966) pdisl

=

30 @ Ndisl

d 2 (kB T ) 3 1 2

e3f

Ad

fi

(13-1)

800


Figure 13-25. Electron mobility vs. temperature for silicon-doped n-type GaN samples of varying amounts of n-type carriers (as measured at RT). These curves represent the case of dominant impurity scattering at lower temperatures. [Data taken from Ng et al. (1998).]

f

I

c

Temperature (K) where d is the distance between the acceptor states along the dislocation line, f is the occupation fraction of the acceptor states which is assumed to be almost constant as a function of temperature, Ndislis the dislocation density, and Ad is the Debye screening length (1 3-2)

Consequently, the electron mobility due to scattering at charged dislocations should increase monotonically with the net carrier concentration. The growth of heteroepitaxial GaN material of superior electrical quality therefore requires exact knowledge of how to handle the amount of stress and the resulting amount of structural (extended) defects in these layers. One example shall be mentioned here. Nakamura et al. (1992 b) after introducing the concept of a low-temperature grown GaN buffer layer, empirically found the maximum electron mobility in the GaN main layer for a buffer layer thickness of about 20 nm. In a related study, Kriiger et al. (1 999 b) showed that the thickness of the low-temperature grown GaN buffer layer determines the amount of residual

stress in the main layer. With increasing buffer layer thickness, the main layer increasingly relaxes via dislocation movement and multiplication. The resulting dislocation density is inversely related to the electron mobility in these layers (Fig. 13-26). For the thickest buffer layer, the main layer is found to be stress-free, but the resulting dislocation density deteriorates the electron mobility.

13.6 Devices Based on 111-Nitrides Despite the severe materials problems of state-of-the-art GaN and related alloys summarized in the preceding sections, GaNbased light emitting diodes are already available from several suppliers, blue semiconductor lasers are being tested for introducing, e.g., in optical data storage applications, and high frequency microwave power amplifier transistors are being developed in several industrial and academic institutions. The following discussion will concentrate on critical defect issues of these devices. A recent review on 111-nitride based devices is given by Pearton et al. (1999).

13.6 Devices Based on Ill-Nitrides

300

I

"

'

2 IO1'

c250

801

Figure 13-26.Room temperature electron Hall mobility and dislocation density of the GaN main layer vs. buffer layer thickness of MOCVD grown GaN on sapphire. The dislocation density was determined by plane view TEM. [Data taken from Kruger et al. (1999b).]

50

0 0

20

40

60

80

100

120

0 100 140

buffer layer thickness [nrn]

13.6.1 Optical Devices: Light Emitting Diodes (LEDs) and Lasers Blue light from GaN-based light emitting diodes was first demonstrated in 1972 by Pankove et al. Due to the absence of p-type material, only rather inefficient M-I-n (metal-insulator-n-semiconductor) structures could be fabricated. The first p/n junction LED was introduced by Nakamura and coworkers in 1994 (Nakamura et al., 1994a) with the first high brightness LED based on an advanced InGaN/AlGaN double heterostructure grown by MOCVD on a GaN buffer layer on sapphire. Amazingly, since that time Nakamura (of Nichia Chemicals) has been able to stay at the forefront or even ahead of the competition in the development of GaN-based optoelectronic devices. This includes the presentation of efficient bluegreen LEDs, e.g., for traffic lights (Nakamura et al., 1994b), the first single quantum well green LEDs (Nakamura et al., 1995), and the first GaN-based blue laser diode (Nakamura et al., 1996). This development has recently been described in Nakamura and Fasol (1997). The short wavelength of the blue LEDs allows the excitation of phosphors for con-

version of the light output to longer wavelengths. Of special interest is the use of organic phosphors emitting in the green and red region, or of suitable inorganic phosphors such as Y,AI,O,, doped with cerium which allow the production of white light (Schlotter et a]., 1997). This technology can be safely expected to revolutionize even standard room illumination. A puzzling question is how these devices can operate efficiently with lifetimes exceeding lo4 h despite a high density of structural defects, including ca. lo8 cm-2 threading dislocations (Ponce et al., 1994; Lester et al., 1995). The high recombination energy of the wide-gap material should result in rapid degradation, as has been experienced for all blue light emitting solid state devices based on II/VI semiconductor technology, see, e.g., Guha et al. (1993). To answer this key question, the first issue to be considered is the high bond strength of GaN compared with that in all I I N I semiconductors. As long as the energy to form a bond is comparable to or even lower than the carrier recombination energy, each recombination process may in principle release enough energy to break a bond. Although in completely defect-free direct

802


band-gap material the recombination energy is efficiently converted to light emission, any defect might serve to convert a sizable fraction of the recombination energy into multiphonon emission or recombinationenhanced defect reactions in semiconductors (Kimerling, 1978). In such a case, even a very low initial defect density might be sufficient to drastically limit the operation lifetime of a light emitting device. In the case of GaN, the bond strength has been estimated as 6 - 9 eV bond-', well above the energy band gap (Newman, 1998). Thus the energy released in recombination-enhanced defect processes cannot come close to the energy required to break up a covalent bond. However, even if the defect density is not expected to be increased by recombination processes, lo8 cm-* of dislocations would be sufficient to kill any radiative output from LEDs based, e.g., on GaAs/AlGaAs heterostructures (Lester et al., 1995). Here the short diffusion length in GaN can be a decisive advantage. Rosner et al. (1999) determined the minority carrier diffusion length in MOCVD grown, high quality GaN

I

.

.

"""I ."."1

.

.......I

.

.......I

as 200 nm. As long as the spacing between dislocations is much larger then the minority carrier diffusion length, light emission can only be reduced by acertain amount, but never completely eliminated. Even under the assumption of dislocations acting as highly active recombination centers, only a cylinder around each dislocation with the radius of the carrier diffusion length will be optically dead. In the case of a 2 0 0 n m diffusion length, a dislocation density of lo8 cm-2 would mean that at most 25% of the material would not participate in emitting light. Further on, the commonly observed inhomogeneous distribution of indium in the InGaN active layer of LEDs and laser diodes can result in carrier localization (Chichibu et al., 1998; Yu et al., 1998; Shapiro et al., 2000), which further decreases the probability of nonradiative recombination at dislocations and other defects (Nakamura, 1999b). This might also explain why the degradation mechanism of InGaNbased heterostructures differs from that of all other 111-V semiconductor opto-electronic devices (Fig. 13-27).

.

...--.I

. ...*

InGaN (low current)

.

InGaN (high current):

I

. . . ,,,.., . . ......, .

-.....I

........ ........, ........ ........, 1

1

Figure 13-27. Degradation of 111-V heterostructures as a function of the dislocation density. [Figure taken from Lester et al., 1995). Data on GaN and

13.7 Outlook

13.6.2 Electronic Devices: Field Effect Transistors (FETs) Similarly as for GaAs-based electronic devices, there is no stable oxide known on GaN that would allow the formation of an inversion layer and thus metal-oxide-semiconductor FETs (MOSFETs), which are so successful in mainstream silicon technology. However, the fact that 111-nitrides form ternary alloys of GaN with aluminum and indium allows the fabrication of high-electron mobility transistors (HEMTs) based on electron conduction in a two dimensional electron gas at the interface of, e.g., AlGaN and GaN (Khan et al., 1993). With this approach, impressive results in the fabrication of microwave power amplifiers have been achieved (Y. E Yu et al., 1997; Nguyen et al., 1998; Sullivan et al., 1998). It is expected that III-nitride based power amplifiers will be able to exceed the power and frequency performance of all competing approaches. However, the high power performance of these devices is still hampered by an effect called current compression at high applied voltages (Kruppa et al., 1995). A similar effect encountered in the development of AlGaAs/GaAs HEMT structures was found to be due to deep level defects. Very recently, a similar conclu-

sion has been reached for AlGaN/GaN HEMTs, too (Nozaki et al., 2000). A very interesting property of AlGaN/ GaN heterostructures is the large piezoelectric effect which results in a spontaneous polarization field at the interface, which is further modified by any strain present in the heterostructures (E. T. Yu et al., 1997). This field can supply the electrons required for the sheet charges of the 2d electron gas even without any chemical doping, but in turn the carrier concentration can depend on the amount of strain and the polarity of the material, including the density of antiphase domain boundaries.

13.7 Outlook This chapter can only indicate why the future of this novel materials system looks so promising. In many laboratories around the world, intensive work is under way for improving the materials quality by reducing the density of structural and electrically active defects, controlling the strain, improving the reproducibility and reliability of devices, and optimizing device design and processing. This is evidenced by the rapidly increasing number of papers in this field

1968 1972 1976 1980 1984 1988 1992 1996 2000

year

803

Figure 13-28. Chronological plot of the number of publications on GaN per year. Data have been extracted from the INSPEC database searching for the keywords GaN, InN, and AIN.

804


(see Fig. 13-28). It can be safely expected that devices based on 111-nitrides will pass the volume of devices based on other IIIN semiconductors, as they open up new areas of applications, such as solid state room illumination, which were not accessible before and which will be in use in everybody’s household. The need for intelligent defect engineering to master this complex material will require further strong research efforts in the fundamental materials science of this unique ceramic semiconductor system and thus provide the basis for sustained support in this field.

13.8 Acknowledgements The authors gratefully acknowledge the interactions and discussions with past and present members of the GaN research team at Lawrence Berkeley National Laboratory, specifically J. W. Ager 111, D. Corlatan, H. Feick, H. Fujii, Y. Kim, R. Klockenbrink, M. Leung, Z. Liliental-Weber, N. Newman, S. Nozaki, P. Perlin, M. Rubin, N. Shapiro, H. Siegle, S. Subramanya, W. Walukiewicz and C. Wetzel. This review was supported by the US Department of Energy under Contract No. DE-AC03-76SF00098.

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INDEX

Index Terms

Links

A A-swirl defects

281

α-fringe technique

400

408

absorption – chalcopyrites

759

– dislocations

436

– germanium

32

– quantum coherence

76

– silicon carbides

686

– solar cells

718

acceptor–hydrogen complexes – deep centers

221

– gallium nitride

787

acceptors – band theory

13

– chalcopyrites

760

– deep centers

170

– dislocations

303


100


666

– transition metals in silicon

604

Acheson growth

665

activation barriers, intrinsic point defects

125

197

687

692 f

activation energy – copper in silicon

614

– dislocations

356

– silicon vacancies

132

active matrix liquid crystal display (AMLCD)

583

adressing, thin films

589

589

This page has been reformatted by Knovel to provide easier navigation.

Index Terms affine transformations, grain boundaries agglomeration

Links 384 280 ff

– copper silicides

630

– native point defects

365

– oxygen defects in silicon

742

– silicon interstitials

218

Aharonov–Bohn interference

70

algebraic structural units, grain boundaries

394

allotaxy

516

aluminum acceptors

666

aluminum arsenide system

457

aluminum–gallium–arsenic alloys

207

212

aluminum gettering

640

647 f

aluminum interstitials, silicon

137

aluminum nitrides

771

– grain boundaries

415

aluminum oxides

532

aluminum–silicon interfaces

518

687

692

523

ammonia – gallium nitride

778

– solar cell silicon

747

amorphous silicon – band theory

63

– hydrogenated

541

– solar cells

750 ff

amorphous/crystalline interfaces

524 ff

amphoteric defects

320

amphoteric dopants

175

anion antisites V(III)(5i3), intrinsic point defects

149

anisotropic mobilities

704

annealing – dislocations

296

– germanium

427

– gettering

640

322

647 f


Index Terms

Links

annealing (Cont.): – hydrogenated a-silicon

571

581

– interfaces

479

501

513

– intrinsic point defects

146

– silicides

508


700

– silicon diffusion

247


741


628

149 ff

160

496

527

annihilation – defects

568

– kink–kink pairs

352

antimony chalcogenides, band theory

39 f

antimony diffusion

262

antisites – chalcopyrites

759

– dislocations

329


125

– point defects

234

applications – band theory

12

– envelope function approximation

61

– gallium nitride

774 ff

– hydrogenated a-silicon

583 ff

Arrhenius law – deep centers

175

– dislocations

315

– interfaces

480


701


609

arsenic

620

12

arsenic doping – grain boundaries

431


556

559


Index Terms

Links

arsenic doping (Cont.): – interfaces

477

arsenic self diffusion

273

a-silicon, band theory

9

asymmetry, band theory

40

athermal diffusion

246

athermal migration, silicon vacancies

132

atom probe field microscopy (FIM)

383

atomic force microscopy (AFM), interfaces

527

atomic spacing, interfaces

457

400

atomic structure – hydrogenated a-silicon

550

– silicide precipitates

625

atomic-resolution Z-contrast, grain boundaries

432

atomistic diffusion Auger recombination average t-matrix approximation axial next nearest neighbor Ising model (ANNI)

241 f 724 51 669

B ball-and-stick model – interfaces

521

– nickel silicides

627

ballistic electron emission microscopy (BEEM)

674

ballistic properties, quantum coherence band–band recombination, solar cells

92 724

band bending – dislocations

296


562

band edge related transitions, gallium nitride

789

band edge states, hydrogenated a-silicon

571

band gap bowing

307

314

56

band gaps – CIS

758

– gallium nitride

774


Index Terms

Links

band gaps (Cont.): – interfaces

455

– monocrystalline compounds

756

– photonic

94


670

– solar cells

718

722

755

band recombination – hydrogenated a-silicon

581

see also: recombination band structures – quantum coherence – silicon carbides band tail states, hydrogenated a-silicon band theory

70 670 ff 553

563

1

bandlike states, silicides

635

barrier height – grain boundaries

422


569


609

Bell–Kaiser model

675

beryllium doping

243

beryllium oxide

146

Bessel functions

34

Bethe–Salpeter equation

34

Bethe lattice approximation

50

bias created defects, hydrogenated a-silicon

582

bicrystal transport experiments

426

265

276

binding energies – deep centers

170


690


343

bipolar transisitors

641

bismuth surfactant technique

781

bleaching, quantum coherence

111


Index Terms Bloch function Bloch states

Links 687 74

Bloch theorem

91

113

5

Bloch waves – hydrogenated a-silicon

552

– interfaces

470

Bohr magneton

181

Bohr radius – deep centers

170

– excitons

33


86


686

Boltzman transport equation (BTE)

98 f

Boltzmann constant – hydrogenated a-silicon

564

– point defects

234

bond breaking, dislocations

338

572

bond orbitals – band theory

24


127

bonding, hydrogenated a-silicon

549

bonds, dislocations

573

338 ff

Born–Oppenheimer approximation

177

Born–von Karmann conditions

5 ff

Born approximation

104

Born effective charge

679

boron acceptors

687

boron doping – extrinsic silicon

615

– gettering

645


555

– interfaces

476

boron-hydrogen complexes

749

559


Index Terms

Links

boron impurities – intrinsic point defects

151

– silicon

726


732

boron interstitials, silicon

137

boron pairs, silicon deep centers

213

boundary conditions – band theory

5 ff

– grain boundary models

396

bowing configurations, interfaces

495

Bragg condition, silicides

512

Bragg mirror microcavities

95

Bravais lattice

297

breakdown properties, interfaces

529

Bremsstrahlung isochromate spectroscopy (BIS)

35

Bridgman process

729

Brillouin scattering, quantum coherence

104

Brillouin zones – band theory

9

– quantum coherence – silicon carbides

22

28

681 ff

687

71 672 ff

brittle-to-ductile temperature, gallium nitride

775

broken bonds, dislocations

335

buffer layers

500

building blocks, intrinsic point defects

125

bulk defects

296

bulk growth

777

506

bulk impurities – grain boundaries

431


555

bulk lattice parameters

488

bulk modulus

75

bulk references, deep centers

197


Index Terms

Links

Burgers vector – dislocations

– gallium nitride – grain boundaries – interfaces

297 ff

311 ff

350 ff

359 ff

326 ff

784 387 ff

402 f

488 f

495 f

508

471

477

721 f

756


627

buried interfaces

455

C cadmium compounds, dislocations

331 ff

cadmium selenide/sulfide, solar cells

756

cadmium telluride, solar cells

718

capture cross section

701

capture–emission process, dislocations

316

capture properties, silicon/germanium

428

carbon doping – gettering

646

– interfaces

480

– silicon

259

carbon impurities – silicon precipitation

283


735

carbon–oxgen interactions, solar cell silicon

743

carbon self-interstitial complexes (CI)

259

carrier–carrier scattering, quantum coherence

104

carrier distributions, quantum coherence

111

carrier dynamics, ultrashort time scales

112

carrier effective masses

704

carrier–lattice coupling

70

741

carrier lifetimes – silicon carbides

706

– solar cells

718

carrier mobilities, gallium nitride

797

carrier properties, quantum coherence

721

98


334 ff

Index Terms carrier recombination, dislocations

Links 296

carriers – grain boundaries

424

– interfaces

500

– point defects

503

236 ff


683


754

cathodoluminescence – dislocations

333

– interfaces

503

cation antisite III(V), intrinsic point defects

151

cationic sites, band theory

9

cavitation, grain boundaries

441

cavity gettering

640

cell periodic wave functions, quantum coherence

74

center-of-mass model, quantum coherence

79

center sites, hydrogenated a-silicon

577

cerium oxide

532

chain like structures, band theory chalcogen series, silicon deep centers chalcogen vacancies, intrinsic point defects

347

647 f

37 202 ff 148

chalcopyrite – band theory

56

– solar cells

758

charge controlled recombination (CCR)

322

charge state controlled metastability

214

charge transport, quantum coherence

70

charged interstitial-substitutional species charged point defects

95 ff

245 235 ff

charge-to-breakdown, interfaces

527

chemical bonding, hydrogenated a-silicon

550

chemical driving force

622

chemical gettering, transition metals in silicon

639

chemical interfaces

455


Index Terms chemical lattice imaging chemical mapping

Links 459 ff 456

chemical vapor deposition (CVD) – interfaces

476


691

– solar cells

718

chirality, grain boundaries

402

723

750

chromium – gallium arsenide indiffusion

243

– silicon deep centers

194

– silicon indiffusiion

613

chromium doping, gallium arsenide

265

223

chromium impurities – silicon carbides

697


733

cleaning techniques, interfaces

526

close Frenkel pairs – intrinsic point defects

148

– silicon

140

– zinc

145

close packing, silicon carbides

665

close spaced sublimation (CSS)

757

clusters – amorphous silicon

65


393

– interfaces

486

cobalt impurities, solar cell silicon

733

cobalt precipitation – silicon deep centers – silicon

222 603 f

622

625

cobalt silicides, interfaces

507

510

523

codeposition, metal-silicon interfaces

507

coherenece, quantum coherent potential approximation (CPA)

69 51 f


Index Terms

Links

coincidence site lattice (CSL)

385

colony growth, copper silicides

630

color pixels, matrix addressed arrays

590

compensation model doping, hydrogenated a-silicon

571

complementary metal oxide semiconductor (CMOS) transistors

477

529

compounds – deep centers

197

– deep dislocation levels

344

– dislocations

357

computational techniques, band theory

19

computer simulation techniques, grain boundaries

395

concentration profiles, zinc in silicon

251

conductances – grain boundaries

422


107

conduction band – band theory

22

– deep centers

170

– dislocations

320

– gallium nitride

789


417


550


72 670 f

681


554

564


756

– silicon

622

configurational coordinate (CC) model

129

confined systems, quantum coherence

94

confinement effects, quantum dots

47

conductivity

conglomeration, silicide precipitates

516

conjugate gradient method, grain boundaries

396


Index Terms constrictions, dislocations

Links 301

contamination – deep centers

171

– silicon

630

– siliconoxide interfaces

525

– dislocations

327

– interfaces

503

continuum excitons contrast reversal, interfaces

79 477

convergent beam electron diffraction (CBED) – gallium nitride

785


400

cooling rates, transition metals in silicon

630

coordination number – band theory

36

– interfaces

521

copper – germanium indiffusion

243

– point defects

234

– silicon indiffusion

241

– stacking faults

642

copper-based ternary compounds

759

copper doping

433

copper impurities, silicon

603 f

copper interstitials

188

copper precipitation, silicon

630

copper silicides, colony growth

630

copper silicon, deep centers

194

coprecipitation, silicon

283

core bond reconstruction, dislocations

264

733

634

222

338 ff

core states – dislocations

334

– interfaces

503


128


Index Terms

Links

correlated electron hole pairs

76

correlation effects, band theory

16

Coulomb energy, grain boundaries

399

Coulomb-enhanced Auger recombination

725

Coulomb-induced nonlinear screening

192

Coulomb interactions – band theory

17

47

– deep centers

170

188

202

– dislocations

296

304

313

340

– interfaces

467


141 86

93

112


78 696

coupling constant – band theory

30

– deep centers

177

– gallium arsenide

415

covalent bonding – band theory

36

– dislocations

346


549

creep test

369

critical resolved shear stress (CRSS)

369

critical thickness, misfit dislocations

490

cryogenic temperatures – gallium nitride

797


126

– silicon

140

crystal field model

180

crystal field splitting

677

crystal growth

695

233 ff

crystal structures – dislocations

297

– interfaces

471

489


Index Terms

Links

crystal structures (Cont.): – silicon carbides crystalline band structures crystalline silicon crystalline solids, Bloch theorem crystalline systems, composition dependence crystalline/amorphous interfaces

665 ff 14 ff 718 ff 7 505 524 ff

cubic structures – gallium nitride

796

– interfaces

489


666

CuInGaSe2 (CIGS)

719

723

CuInSe2 (CIS)

719

758

current-created defects, hydrogenated a-silicon

580

762

current–voltage characteristics – grain boundaries

423

– solar cells

720

Czochralski silicon

262

– agglomeration

281

– deep centers

216

– dislocations

302

– gettering

641


431

– solar cells

718

727 ff

D d orbitals

21

D-swirl defects

281

δ-function, interfaces

465

damage, silicon diffusion

247

dangling bonds – band theory

47

– deep centers

186

204

– dislocations

299

308 f


364

417 f


Index Terms

Links

dangling bonds (Cont.): – hydrogenated a-silicon

558 f

– interfaces

503

– silicon

134

de Broglie wave length

89

Debye–Hückel approach

88

574

755

Debye frequency – dislocations

359


611

– interfaces

496

Debye screening

305

deep centers

167


314

694 ff

deep double donors

149

deep electron levels, dislocations

336

deep level transient spectroscopy (DLTS) – band theory

47

– deep centers

173

205

216

– dislocations

296 ff

303

315

696

700 ff


425


560

– interfaces

503

– silicides

636


692

– silicon

131


609

deep levels – dislocations

343 f

– gallium nitride – silicides

787 635 f


687

defect compensation model doping

571

defect concentration, doping

558

defect-molecule model, deep centers

183


Index Terms defect reaction kinetics, hydrogenated a-silicon

Links 576

defect states – hydrogenated a-silicon – interfaces

558 f

568

580

503

defects – chalcopyrites

760

– deep centers

170

– dislocations

296

– gallium nitride

777

783 ff


384

390

– interfaces

495

– intrinsic

121

– point

231


96 666 ff

681

700

– solar cells

725

728 ff

741

deformation-induced luminescence

333

deformation-induced point defects

302 ff

deformation modeling, grain boundaries

443

deformation scattering

103

deformations – dislocations

296


435

deformed bonds, dislocations

335

365 ff

degeneracy – band theory

12

– deep centers

174

202


677

687

degradation, hot electron

529

dehybridization, deep centers

193

delimiting grain boundaries

403

delocalization, band theory

40

dendritic web growth (D-WEB)

738 f


Index Terms

Links

density functional theory (DFT) – silicon carbides – quantum coherence

668 72

density of states – amorphous semiconductors

44


552

– silicon

427

denuded zone formation

640

depletion effects, hydrogenated a-silicon

547

depletion layers

477

deposition techniques – CIS

762

– gallium nitride

775 ff

– solar cells

718

deuterium concentration profiles

579

723

devices – gallium nitride – hydrogenated a-silicon – solar cells

800 583 ff 719 f

diamond structure – dislocations

297

– germanium

411


414


549


141


73


603

dichromatic complex (dec), grain boundaries

387

dielectric constants, oxides

532

dielectric polarization

74 f

dielectric properties, band theory

346

65

dielectric response, nonlinear

110

dielectrics

524

diffraction pattern

460


Index Terms

Links

diffraction peaks

455

diffusion

231


422


558


125

– kinks

356


700


748


603

diffusion-induced grain boundary migration (DIGM)

441

diffusion mode, segregation gettering

651

459

577

609 f

615 f

388

395

diffusivities – grain boundaries

441

– group IV elements

259

– interfaces

479

– point defects

242


249

diffusivity enhancement

255

dimensionality – band theory

57


89 ff

diodes – gallium nitride

801


583

– interfaces

522

dipping

752

disclinations, grain boundaries

395

discrete excitons

79

discrete states, band theory

5f

dislocation climb – gallium arsenide

282


631

dislocation–grain boundary interactions

436

dislocation model

383


Index Terms

Links

dislocation reduction techniques

500

dislocations

291

– gallium nitride

776

784

– interfaces

484

488


508

– point defects

239


668


730


627

disordered alloys disordered systems disordering, gallium arsenide superlattices

493

51 ff 9 266

273

dispersion – phonons

679


76

– s band

38

– silicon dislocations

309

dispersive hydrogen diffusion

558

displacement shift complete (DSC) dislocations

403

displays

590

dissimilar layers, interfaces

455

82

419

dissociation – dislocations

350


548

distortions – grain boundaries

418

– interfaces

485

divacancies

125

Dodson–Tsao model

494

donor–acceptor codoping

788

504

donor–acceptor pairs (DAPs) – gallium nitride

794


692


213


Index Terms donor–host bond breaking

Links 211

donors – band theory

12

– chalcopyrites

760

– deep centers

170

– dislocations

303


557


149


100


666

197

204

687 f

dopants – diffusion

233 ff

– dislocations

303

– extrinsic silicon

615

– gallium nitride

777


431


247

252 f

319 f

324


265

268

– gallium nitride

787


422

431


546

554 ff

571

– interfaces

474

– kinks

351

– polycrystalline compounds

756

673

699

786

doping – dislocations


90


665

double junctions, grain boundaries

421

double kinks, dislocations

359

double layer stacking sequences

665

double sonors, intrinsic point defects

149

double well potential doublet splitting

362 ff

6 695


Index Terms

Links

drift mobility, hydrogenated a-silicon

566

drip-controlled method, solar cell silicon

729

dry cleaning, interfaces

526

DX states – aluminum gallium arsenic alloy

207

– gallium nitride

787

dye sensitized titanium dioxide

765

dynamic properties, grain boundaries

424

Dyson equation

53

E edge defined film fed growth (EFG)

737

edge dislocations

300

effective mass theory (EMT) – band theory

5

10 f

– deep centers

170

201


73

86

– silicon

63

efficiency limiting defects

29

741

eigenstates – band theory

18


74


679

eight folded types, silicide interfaces

521

elastic distortions, interfaces

485

elastic modulus

611

elastic strain

611

electric dipole resonance (EDSR)

326

electrical activity, grain boundaries

424

electrical behavior, solar cell silicon

740

electrical characteristics, photodiodes

585

electrical conductivity, hydrogenated a-silicon

564

electrical level position, silicon vacancies

130

electrical measurements, point defects

313

27


Index Terms

Links

electrical properties – doped silicon

221

– gallium nitride

797


417 ff

– misfit dislocations

502

– silicide precipitates

635

electrical transport, quantum coherence

71

electrodepostion, solar cell silicon

752

electroluminescence

201

electromagnetic casting techniques (EMC)

729

97

electron beam induced current (EBIC) – dislocations

303


426

– interfaces

503


728

electron cyclotron resonance (ECR)

780

electron densitiy, band theory

314

320

7

electron diffraction, grain boundaries

408

electron drift mobilities, hydrogenated a-silicon

567

electron energy loss spectroscopy (EELS) – gallium nitride

786


383

electron–hole excitations, band theory

31

34 f

electron–hole pairs – photodiodes

586


78

– solar cells

718

electron holography

475

electron hopping

303

electron irradiation, intrinsic point defects

126

electron masses

672

electron nuclear double resonance (ENDOR)

693

721

149

electron occupancies – interstitials

159


327

Index Terms

Links

electron occupancies (Cont.): – vacancies

157

electron paramagnetic resonance (EPR) – deep centers

172

– diffusion

216

233 ff

– dislocations

299

308 f

– interfaces

503


126

143

– kinks

340

357


609

electron phonon coupling – deep centers

199


102

electron–photon interactions, deep centers

178

electron recombination

424

electron self-energy operator

17

electron spin resonance (ESR) – band theory

47


557


693

electronic band structures

71

electronic confinement

91

electronic devices, gallium nitride

701

803

electronic levels – dislocations

334 ff


687

electronic properties – cadmium telluride

757

– chalcopyrites

759

– CIS/CIGS

763

– dislocations

302 ff

– gallium nitride

775


547

– metal-semiconductor interfaces

520


Index Terms

Links

electronic properties (Cont.): – monocrystalline compounds

756


730

739

electronic states – grain boundaries – hydrogenated a-silicon

417 552 ff

570

electronic structures – deep centers

180 ff


213


127

– silicon

619


603

electronic transitions, deep centers

174 ff

electronic transport – hydrogenated a-silicon – solar cell silicon

564 ff 753

electron-nuclear double resonance (ENDOR) – deep centers – intrinsic point defects

172

192

126 ff

152

electroplastic effect, dislocations

333

embrittlement

435

emission properties, silicon/germanium

428

emitter push effect phosphorus diffusion

256

216

empirical pseudopotential method (EMP) – silicon

63

– band theory

19

– zincblende structures

26

empirical tight binding (ETB), silicon

63

energetic structural units energy conversion, solar cells

393 719 ff

energy distributions, grain boundaries

421

energy gaps, silicides

635

energy levels, band theory energy loss near-edge spectroscopy (ELNES)

22 ff

5f 400


Index Terms energy minimization method, grain boundaries

Links 396

energy spacing, donor impurities

13

engineered band structures

70

enhanced diffusion

254

entropy terms, deep centers

175

envelope function approximation

5

environmental dependent interatomic potential (EDIP)

341

epilayer shear modulus

484

epitaxial deposition, gallium nitride

778

epitaxial elemental metals

518

epitaxial growth, silicon carbides

670

epitaxial lateral overgrowth (ELOG)

781

epitaxial silicides

472

epitaxial wafers

645

epitaxy

502

equilibrum conditions, point defects

10 f

59

523

520

234 ff

erbium impurities

695

etch pitting

347

eutectic temperature, silicon

630

evaporation – solar cells

718


639

excess stress, interfaces

490

excitation spectra, gaps

85

723

excited electrons/holes, solar cells

724

excited states, deep centers

199

exciton band gaps, silicon carbides

670

678

exciton–photon coupling

80

94

exciton screening

87

excitonic exchange splitting

64

excitonic recombination – interfaces

457

see also: recombination excitonic spectrum, calculations

34


Index Terms

Links

excitons – band theory

31

– gallium nitride

791

– interfaces

474

– quantum coherence – silicon carbides expansion coefficient, gallium nitride

72

78

686 ff

707

104

776

experimental techniques – dislocations

302 ff

– grain boundaries – interfaces

399

424

455 f

extended defects – dislocations

303

– gallium nitride – silicides

315

783 f 637

extended X-ray fine structure (EXAFS) – band theory

54

– deep centers

172

external gettering, transition metals in silicon

639

646

233 ff

296

extrinsic defects extrinsic electrical activity, grain boundaries

428

F Fabry–Perot oscillations Fabry–Perot resonator

680 94

face-centered structures – band theory

25

– interfaces

489

facet bar density, interfaces

509 f

fast diffusing impurities

171

– silicon

625

favored grain boundaries

403

Fermi level – band theory

7

– cadmium tellurides

757


Index Terms

Links

Fermi level (Cont.): – chalcopyrites

760

– deep centers

174

180

– dislocations

296

305 f

320


274

– gettering

650


422


558

564

571


151

– point defects

235 ff


71

76

– silicides

636

– silicon

134


604

615

73

95

fermions Fick diffusion

240 ff

field-effect transistors (FET)

803

field enhancement

240

film cracking

776

film growth, hydrogenated a-silicon

547

final start/right hand (FS/RH) convention

388

finite area epitaxy, interfaces

502

first order Raman scattering, silicon carbides

682

floated zone silicon

262


427

– precipitation

633

– solar cells

718

floating bond hypothesis, hydrogenated a-silicon

558

flow patterns, solar cell silicon

728

fluctuations, dislocations

317

fluorite structures, silicides

510

Foker–Planck equation

745

folded coordinates, hydrogenated a-silicon

556

folded types, silicide interfaces

521

727

365


Index Terms

Links

force balance model

490

foreign point defects

233 ff

formation – dislocations

342

– kinks

350

Fourier transform – band theory

11 ff

19

457

475


680

698


735

– interfaces

28

Fourier transform infrared spectrometry (FTIR)

Franck–Condon transitions

176 ff

Franck–Hertz experiment

103

Frank–Bilby formula

388

Frank–Read source

495

Frank–Turnbull mechanism – diffusion

243

250


272

276

Frank model

668

Frank partials – dislocations

328

– gettering

642

Franz–Keldysh effect

330

freezing

352

366

569

Frenkel pairs – intrinsic point defects

148

– silicon

140

– zinc

145

frequency – grain boundaries

424

– interfaces

457

– phonons

681

friction stresses

352

Friedel oscillations

669

496


Index Terms

Links

fringes

440

Fröhlich scattering

103

fundamentals, silicon carbides

661

G G-W approximation

17

Γ-point, band theory

30

gadolinium oxide

532

gallium acceptors

687

692

– band theory

27

34

– deep centers

197

gallium arsenide

– diffusion

233 ff

265 ff

– dislocations

329 f

– DX centers

210


415

– interfaces

457

495

70

73

– solar cells

718

721

gallium nitride

771


415


764

gallium phosphide – deep centers

183


149

gallium self-diffusion

266

197

gap states – deep centers

187


560

575

84

94

– quantum coherence gate dielectrics

529 f

geometries – dislocations

297 ff


385

germanium – band theory

12

34


Index Terms

Links

germanium (Cont.): – diffusion

264

– dislocations

313

326 ff

362


401

411

427


141

– partial migration

342


73

– Shockley partials

440

germanium bicrystals

443

germanium indiffusion

259

germanium monochalcogenides

39

germanium–silicon alloys

646

germanium–silicon/selenium, interfaces

486

gettering – dislocations

371


746


603

639 ff

Gibbs free energy – deep centers

174

– point defects

234

– silicon

622

glass transition temperature, hydrogenated a-silicon

570

glasses, silicon oxides

49

glide planes – dislocations

331

– interfaces

488

glide-sets, dislocations

341 f

297 ff

335


196

221 ff


243

249 f

346

gold

gold impurities

603 f

graded composition structures

501

gradient freeze process, solar cell silicon

729


369

Index Terms

Links

grain boundaries

377


727

730

grain size – gallium nitride

781


734

graphite, solar cells

729

738

34

187

Green function Greiner–Gibbons pair diffusion group II–VI compounds

274 73

group II–VI semiconductors

127

group III–nitrogen alloy systems

771

143

147

group III–V compounds – deep centers

197


280


127

group III–V semiconductors – epitaxialmetallic compounds

519


149

group III atom vacancies

152

group III dopants

247

group III sublattices

458

156

group IV elements – gallium arsenide

273


73


259

group IV semiconductors

141

group V dopants

247

group V elements

273

group V sublattices

153

155

growth – band theory

57

– gallium nitride

777


546

– interfaces

527

561


155

Index Terms

Links

growth (Cont.): – silicon carbides

670


727

Gunn effect, quantum coherence

112

H halides

778

Hall effect – deep centers

173

216

– dislocations

303

313 ff

329

– gallium nitride

797

691

705


564 f


687

Hall free-carrier concentration

208

Ham theory

642

Hamiltonians – band theory

14

– deep centers

181


554

– quantum coherence Hanle effect

82

96

106

Hartree–Fock approximation – band theory

14 f


72

113

Haynes rule – gallium nitride

794


691

heat exchange method, solar cell silicon

729

heavy holes

74

heteroepitaxial deposition heteroepitaxy

775 483 ff

heterogeneous nucleation, interfaces

497

heterogeneous precipitation, transition metals in silicon

633

heterointerfaces

460

heterojunction bipolar transistors (HBTs)

492


Index Terms

Links

heterojunctions, deep centers

198

heterostructures, interfaces

483

heterovalent systems, quantum coherence

91

hexagonal structures – gallium nitride

777


666

high-efficiency materials, solar cells

764

high-electron mobility structures, gallium nitride

774

high-field orthogonal conductance

423

high-field transport, quantum coherence

108 f

high-quality graphite

729

high-quality silicon

718

726

– dislocations

302

355


382

389

– interfaces

520

high-resolution electron microscopy (HREM)

400

high-resolution transmission electron microscopy (HRTEM) – interfaces

459

– silicon

626


735


631

high spin–low spin ordering

187

high stress–low temperature deformation

322

high-temperature properties, transition elements in silicon

597

highest occupied orbital (HOMO), silicon Hirth–Lothe theory Hohenberg–Kohn theorem

63 356 ff 16

hole capture

151

hole drift mobilities

567

hole recombination – grain boundaries

424

see also: recombination holes, quantum coherence Holz reflections

74 415


Index Terms homogeneous nucleation, interfaces

Links 497

hopping – dislocations

303


552

Hornstra model

393

host atoms

125

host lattice

505

host matrix, deep centers

170

hot carriers, quantum coherence

100

hot-electron degradation, interfaces

529

Huang–Rhys factor

177

108

Hubbard correlation energy – deep centers

180

– silicon

130

Hubbard splitting

340

Hull model

494

192

196

Hund’s rule – deep centers

173

195


121

127 f

hydride vapor phase epitaxy (HVPE)

778

hydrogen diffusion

263

– a-silicon

577

hydrogen–glass model, a-silicon

564

hydrogen impurity complexes

749

hydrogen passivation – solar cells

718

– thermal donors

219

hydrogenated amorphous silicon

541

hydrogenic acceptor states hydrogenic energy levels, gallium nitride hydrogenic impurities

747

13 787 5

hydrostatic strain

777

hyperfine interactions, hydrogenated a-silicon

557

hyperfine structure (HFS), dislocations

308


Index Terms

Links

I illumination, solar cells

723

illumination-created defects

580

image sensors, hydrogenated a-silicon

591

impurities – band theory

12

– deep centers

170

– diffusion

233 ff

– dislocations

340 f


616


274

– gallium nitride

777


555


151


786

96 694 ff


204

– silicon

603

impurity segregation

431

222

indiffusion – gallium arsenide

271

– interfaces

479

– point defects

238

indium–copper antisites

761

indium doping

756

indium gallium arsenide, interfaces

460

indium nitrides

771

indium phosphides – dislocations

369

– interfaces

460


149

– solar cells

764

indium tin oxide (ITO)

724

inequivalent sites, silicon carbides

666


Index Terms

Links

infrared absorption, silicon carbides

691

infrared transmission, silicon carbides

679

ingot growth technologies, solar cell silicon

729

inhomogeneous doping

474

injection gettering

649

interactions – grain boundaries

398


499

– vacancies–defects

133

interband absorption spectra

125

83 ff

interdiffusion – gallium arsenide superlattices

270

– interfaces

480

interface roughness

786

interfaces

453

– hydrogenated a-silicon interfacial electronic structures, grain boundaries interference, quantum coherence

418 70 513

intermediate grain boundaries

405

intermetallic compounds

524

intermixing, interfaces

481

internal transitions, deep centers

487

561 f

interlayer-mediated epitaxy

internal gettering

484

639 f 199

interstitial impurities – diffusion

233 ff

– gettering

641

233 f

interstitial-substitutional mechanisms – diffusion

243

– gallium arsenide interstitial model, thermal donors

273 ff 218

interstitials – copper in silicon

194

– deep centers

182

188


Index Terms

Links

interstitials (Cont.): – diffusion

242


577

– interfaces

480


125

– silicon

255

152

159

135 ff

– zinc

145

intra-atomic parameters, band theory

59

intrinsic defects – dislocations

296


700

intrinsic electrical activity, grain boundaries

428

intrinsic excitons

686 ff

intrinsic gallium arsenide

265

intrinsic point defects

121

– dislocations

366


728

intrinsic silicon

233 ff

604 ff

inversion domain boundaries (IDB)

384

– aluminum nitrides

415

– gallium nitride

415

ion-assisted deposition, solar cell silicon

752

ion beam-induced current (IBIC)

426

783

ion implantation – interfaces

479 f

– point defects

238


699

– silicon

261

ion scattering, interfaces

456

ionization – acceptors

692

– deep centers

170

– nitrogen donors

688


79

174

194

100


217

Index Terms

Links

ionization (Cont.): – silicon deep centers

222

iron – diffusion in silicon

241

– gallium arsenide indiffusion

243

– point defects

234


190

– stacking faults

642

iron–boron pairs

215

iron doping

434

223

iron impurities – silicon

603 f


733

iron precipitation, silicon

633

island growth – gallium nitride

783

– interfaces

486

isochronal annealing, dislocations

323

isolated dislocation treatments

334

isolated transition metal impurities

222

isolated vacancies

127

isomer shift, extrinsic silicon

616

isostructural crystalline systems isotropic band structure

455 f

509

133

483 ff

686

J Jahn–Teller distortions jogs

128 f

189

196

301

joint densities, band theory Joule heating

32 108

junctions – grain boundaries

421


583

– interfaces

476


90


Index Terms

Links

junctions (Cont.): – solar cells

719

730

K k-p description

29

Kane derivation

13

Keating model – dislocations

338


402

Kerr effect

112

412

kick-out mechanism – diffusion

243


271

– silicon gettering

653

kinetic relaxation models, interfaces

493

250

kinetics – hydrogenated a-silicon

576


493

kinks – dislocations

340 ff


554

– interfaces

499

Krönig–Penney model

6

L Langer–Heinrich rule

697

lap top displays

590

large lattice relaxation (LLR)

207

lasers

774

782

801

lattice constants – gallium nitride

776

– interfaces

471


756


667

488

679


Index Terms

Links

lattice defects – dislocations

303


247

– solar cells

720

726

741

see also: defects, ponit defects lattice match, quantum coherence lattice-matched isostructural systems lattice mismatch, interfaces

90 457 ff 455 f

483 ff

– deep centers

194

207


560

– silicon

129

lattice relaxation

see also: relaxation lattice sites – interfaces

457

– point defects

242

lattice structures – dislocations

297


383


73

96

layers – band theory

38

– interfaces

455

500

leakage, solar cells

721

740

leakage-limited yield

641

Lely crystals

681

length scales, interfaces

472

lens aberrations

460

lifetimes – silicon carbides

706

– solar cells

718

Lifson–Warshel potential

339

721

light beam induced current (LBIC) – grain boundaries

426


Index Terms

Links

light beam induced current (LBIC) (Cont.): – solar cell silicon

730

light emitting diodes (LED) – deep centers

171

– gallium nitride

774

– interfaces

484

light holes, quantum coherence

801

74

Lindhard dielectric function

111

line tension stress

490

linear combination of atomic orbitals (LCAO) – band theory

20

– dislocations

339


398

linear optics

75

linearized Muffin–Tin orbital in Atomic spheres approximation (LMTO-ASA) linewidth, interfaces Liouville theorem

187 474 99

liquid encapsulation (LEC)

152

liquid–liquid extraction, solar cells

726

liquid phase epitaxy (LPE)

718

751

lithium – diffusion in silicon

241

– point defects

234

local densities of states (LDOS)

186

local density approximation (LDA) – band theory

16 f


668

– silicon

63

local imaging, interfaces

462

local vibrational mode (LVM) – intrinsic point defects

126 ff

– silicon

135

localization–delocalization, deep centers

172

151


Index Terms localized energy levels, gallium nitride localized orbital expansion

Links 787 12

localized states – dislocations

318


552 ff


681

Lomer dislocations

326

lone pair electrons

37

Longini mechanism – diffusion

243 f


276

longitudinal conductance, grain boundaries

423

low-field orthogonal conductance

422

low-pressure CVD

754

low-temperature photoluminescence (LTPL) lowest unoccupied orbital (LUMO) Ludwig–Woodbury model

681 f

686

689 f

63 183

186

luminescence – deep centers

201

– dislocations

333

– gallium nitride

795


559

– interfaces

472

M macroscopic structures, interfaces

472

magnetic circular dichroism absorption (MCDA)

697

manganese – gallium arsenide indiffusion

243


222

manganese impurities – silicon carbides – silicon

699 603 f


733

mass spectroscopy, deep centers

172


702 f

Index Terms mass transport, intrinsic point defects

Links 125

material properties – hydrogenated amorphous silicon

541


724

Mathiot model

218

matrix-addressed arrays, hydrogenated a-silicon

589

Matthews–Blakeslee model

490

Matthiessen rule

101

Maxwell equations

728

504

81

mechanical stability, gallium nitride

775

mechanical stress

365

medical imaging

590

medium energy ion scattering (MEIS)

455

521

531

melting points – gallium nitride

775


611

mesoscale structures – interfaces

472

– quantum coherence mesotaxy

70

89

516

metal atom contaminations – deep centers

171

– dislocations

327

metal-boron pair impurities

732

metal interstitials

152

metal–organic chemical vapor deposition (MOCVD)

266

metal–organic vapor phase epitaxy (MOVPE)

778

metal–semiconductor structures, epitaxial

505

metal silicides, precipitation

621

metal–silicon compounds, interfaces

507

metallic compounds, epitaxial

519

metallic impurities, silicon

603

metastability – hydrogenated a-silicon

568

578


Index Terms

Links

metastability (Cont.): – silicon deep centers

207

metric minimization methods, grain boundaries

396

Meyer–Neldel rule

565

microcavities

71

microcontamination, siliconoxide interfaces

526

microcrystalline films

752

214

94

microdefects – hydrogenated a-silicon

558


633

see also: defects microstructures – DX centers

210

– interfaces

459


739

see also: structures microtwinning – dislocations

300


509

microwave conductivity (MWC)

324

migration – dislocations

342


441

– interfaces

480


125


247


135


132


748


609

– zinc vacancies

145

minimal basis set

20

minimum energy approximation (MEA)

160

41

306


Index Terms misfit dislocations – interfaces

Links 370 488 ff

mobilities – dislocations

346

– gallium nitride

797


441


552

– kinks

350


704

MOD DRAM, deep centers

171

modulation doped field effect transistors (MODFETs)

492

molecular beam allotaxy (MBA)

518

358 ff

566

molecular beam epitaxy (MBE) – band theory

57

– carbon diffusion

260


266

– gallium nitride

778 ff

– interfaces

507


153

– solar cells

718

molecular coherent potential approximation (MCPA) molecular dynamics, grain boundaries

54 396

molecular level, selenium/tellurium

37

molecular model, band theory

24

molecular structure, hydrogenated a-silicon

549

molybdenum impurities, silicon carbides

698

momentum, quantum coherence monochrome pixels, matrix addressed arrays

77

98

590

monocrystalline silicon – defect structures

741

– solar cells

726 ff

monolayers

461

Monte Carlo methods

396

Moore law

525

471

507


Index Terms

Links

MOSFET – interfaces

524


107

Mössbauer spectroscopy – band theory

40

– deep centers

172


616

Mott–Hubbard correlation energy

180

Mott bonding

323

Mott law

564

Mott rule (8-N)

551

multicrystalline silicon

718

multi-electron defects

604

multijunction cells, hydrogenated a-silicon

583

multilayers

481

multiphonon mechanism

130

multiple point defects

235

multiple scattering

460

multiplet representation indices, deep centers

199

192

729

744

491

multiplication mechanisms – dislocations

375 ff

– interfaces

495

N n-p-n double junctions

421

n-type doping – gallium arsenide

265

– gallium nitride

787

– point defects

238

252


665

700

n-type germanium

328

n-type silicon

320 f

nanopipes, gallium nitride nanostructures, quantum coherence

326

362

786 70


Index Terms

Links

native defects – chalcopyrites

760

– gallium nitride

777


558

native point defects

233 ff

– agglomeration

281

– dislocations

365

– interfaces

480

near-field scanning optical microscopy (NSOM)

456

nearest neighbors, band theory

474

59

neutral defects, hydrogenated a-silicon

563

neutron activation analysis (NAA)

172

nickel – silicon deep centers

195

– stacking faults

642

nickel doping

434

nickel impurities

603 f

nickel oxide

416

nickel precipitation

625

nickel silicides

507

nitridation, silicon diffusion

259

222

733

634

nitrogen – gallium phosphides

183

– silicon silicon deep centers

206

nitrogen desorption

778

nitrogen distribution

530

nitrogen donors

666

NMOS devices

477

noble metals, silicon diffusion

247

noise, interfaces

457

noncrystalline semiconductors, band theory

44 f

687

462 f

nonequilibrum conditions – dopant diffusion – point defects

256 237 ff


Index Terms

Links

nonlinear optics

108

nonlinear screening

192

normalized sun spectrum (AM1.5) nuclear magneton

721 f 181

nucleation – dislocations

361

– gallium nitride

783

– interfaces

486

– silicon precipitation

283


734


631

numerical computations, band theory numerical simulations, oxygen precipitation

365

493

497

751

46 744

O obstacles theory

359

occupancy level, deep centers

175

Ohm’s law

100

ohmic contacts

719

O-lattice

109

383 f

one-electron approximation, quantum coherence one-electron molecular orbital model

72 128

optical absorption – amorphous semiconductors

553

– dislocations

324


673

optical deformation scattering

103

optical detection electron paramagnetic resonance (ODEPR)

126 ff

560

143

optical detection electron-nuclear double resonance (ODENDOR) optical devices

126 ff 801

optical properties – band theory

31

– gallium nitride – interfaces

789 f 472


Index Terms

Links

optical properties (Cont.): – quantum coherence

70 ff


753

optical recombination, silicon carbides

695

optically detected cyclotron resonance (ODCR)

691

92

704

optically detected magnetic resonance (ODMR) – deep centers

172

– gallium nitride

795


693

orbital degeneracy – deep centers

181


157

orbitals – band theory

15


21 f

550


74

– selenium/tellurium

38


127

orthogonal conductances

422

Ostwald ripening

628

Ourmazd–Schröter–Bourret (OSB) model

217

632

outdififusion – carbon

260


271

overshot effect, quantum coherence

112

oxidation – hydrogenated a-silicon

562


259

oxidation-induced stacking faults (OSF) – diffusion

253


729

oxide-mediated epitaxy (OME)

514

oxygen, silicon deep centers

206

oxygen clusters, solar cell silicon

744


Index Terms

Links

oxygen diffusion

263

oxygen doping

432

oxygen impurities – silicon precipitation

283


735

oxygen-induced defects

741

oxygen precipitation gettering

640

oxygen zinc/cadmium compounds

331 f

P p-channel transistor

477

p-doped silicon

732

p-i-n diode

583

p-i-n solar cell

755

p-n junctions – interfaces

476

– quantum coherence – solar cells

90 719

porbitals

730

740

21

+

p/p gettering

645

p state splitting p-type diodes, interfaces

12 522

p-type dopants – gallium arsenide

265

– point defects

238

252

p-type doping – gallium nitride

787

– polycrystalline compounds

756


665

700

p-type germanium, dislocations

313

362

p-type silicon, dislocations

313

347

π-bonding

414

packing, silicon carbides

665

pair excitations pairing, silicon deep centers

362

79 213


Index Terms

Links

palladium

196

palladium contacts

675

palladium impurities

603 f

parabolic band structure

686

partial migration, silicon/germanium

342

partial mobilities, dislocations

352

221 ff

630

partials – dislocations

298

366

see also: Frank partials Patel effect

364

patterned epitaxy, interfaces

502

Pauli principle – band theory

15


89

Peach–Köhler force

350

111

Peierls potential – dislocations

346 ff

– interfaces

491

pendellösung effect

460

356 ff

468

periodicity – grain boundaries

396

– interfaces

468


125

pertubation theory – band model

10

– grain boundaries – quantum coherence

398 80


679

phase diagrams, gallium arsenides

236

phase transformation, interfaces

523

phenomenological model, deep centers

29

180 ff

phonons – quantum coherence – silicon carbides

102 679 ff


Index Terms

Links

phonons (Cont.): – solar cells

724

phophorus donors

687

phophorus indiffusion profiles

256

phophorus silicate glas (PSG)

649

phosphorus diffusion gettering

640

691 f

647 f

phosphorus doping – extrinsic silicon

615


555

photoconductivity

707

photodiodes

584

559

photoluminescence – deep centers

201

– dislocations

322

– gallium nitride

779

789

– interfaces

461

503

photoluminescence excitation spectroscopy (PLE)

473

photoluminescence spectroscopy

679

photonic band gaps

94

photons

718

photoplastic effect (PPE)

333

photoreflectivity

792

photothermal ionization spectroscopy (PTIS)

172

photovoltaic operations, p-i-n diodes

585

photovoltaic properties, solar cells

792

354

719 ff

physical properties – deep centers

170


441


667

– silicon

204

603 ff


728

piezoelectric effect

777

741

pinholes – gallium nitride

785


Index Terms

Links

pinholes (Cont.): – silicides

509 f

pinning

198

pitting

347

pixels, matrix addressed arrays

589

planar defects

414

plane wave expansion, band theory

19

plasma enhanced chemical vapor deposition (PECVD) – hydrogenated a-silicon

546

563


747

750

plasma spraying

752

plasmons

104

plastic deformation – dislocations

296


149

– silicon

641

plastic flow

494

plastic relaxation, interfaces

485

plate like defects

742

329

365 ff

platinum – silicon deep centers

196

– silicon gettering

653

– silicon indifusion

243

platinum contacts

675

platinum impurities

477

point defect–droplet transitions

604

point defects

231

– dislocations

296

– electrical measurements

313

– gallium nitride

777


555

– intrinsic

249 f

603 f

PMOS, interfaces

– interfaces

221 ff

529

302 ff

357

787

479 f 121


365

Index Terms

Links

point defects (Cont.): – silicon carbides

666

– solar cells

725

point-ion crystal field model

180

728

Poisson ratio – grain boundaries

390

– interfaces

483

polar optical scattering

103

polaritons

80 f

polarization – quantum coherence

74 f


682

polaron problem

103

polaron structure

143

polycrystalline films

753

polycrystalline silicon – grain boundaries – solar cells

428 726 ff

polycrystalline thin films

750

756 ff

polysilicon

525

647

polysilicon thin-film transistors (TFTs)

424

polytypes, silicon carbides Poole-Frenkel effect porous silicon

665 ff 697 63

positron annihilation spectroscopy (PAS)

701

potential barriers, grain boundaries

421

potential drop, interfaces

477

precipitation

231 f

– dislocations

365

– grain boundary-induced

431

– segregation gettering

654

– silicides

516

– silicon

624


736

280

742


Index Terms

Links

precipitation (Cont.): – transition metals in silicon

603

precursors, solar cell silicon

728

pregate oxidation surface preparation

526

pressure, point defects

237

primary dislocation network

388

prismatic punching

641

621 ff

processing – polycrystalline compounds

757

– silicon oxide

525

propagation – band theory

7

– kinks

359


497

proximity gettering

639

pseudobinary alloys

54

pseudocontinuum pseudopotential method

7 17

PTIS, deep centers

202

pyroelectric effect, gallium nitride

777

Q quadrupole splitting qualitative features, band theory

616 57

quantitative chemical maps

466

QUANTITEM, interfaces

456

quantum coherence

466 ff

69

quantum dots – band theory

47

– interfaces

485


70

quantum size effects

70

quantum well LCAO model

90 f

339

quantum wells – band theory

58 This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

quantum wells (Cont.): – interfaces

462


70

90 f

quantum wires

70

90 f

quartz

729

quasimolecular dynamics algorithm

396

quenching – copper silicides

637


571


690

R radial distribution function (RDF)

549

radiation damage, intrinsic point defects

148

radiation defects

310

radiation-enhanced dislocation glide (REDG)

331

radiation-induced self interstitials

246

radiative recombination

84

– deep centers

204

– dislocations

327


355

204

84


686

– solar cells

724

radio frequency diode plasma reactor

546

radio frequency-generated plasmas

780

Raman scattering – interfaces

456


104


679 f

Ramsdell notation

666

random phase approximation (RPA)

111

rapid chemical annealing, (RCA)

318

rapid chemical cleaning (RCC)

526

rapid quenching

571


Index Terms

Links

rapid thermal annealing (RTA) – interfaces

513


247

rapid thermal oxidation (RTO)

527

rare earth impurities

699

rare earth monopnictides

520

reaction kinetics, hydrogenated a-silicon

576

Read model

303

rebonding, dislocations

299

reciprocal lattice

317

8

recombination – band theory

65

– deep centers

171

204

– dislocations

296

303

327

– gallium nitride

794


424

– interfaces

504

686 ff

695

707

– solar cells

718

724

– vacancy-interstital

126

recombination-enhanced defect formation

774

recombination-enhanced diffusion

246

recombination-enhanced migration

130

recombination-enhanced reaction rate

580

reconstruction, dislocations

296

recovery properties, grain boundaries

442

recrystallization

718

reflection, silicon carbides

679

rehybridization, grain boundaries

419

– radiative see: radiative recombination – silicides

636


139

338 ff

relaxation – chemical interfaces – deep centers

479 176 f

194

207


Index Terms

Links

relaxation (Cont.): – dislocations

309

– gallium nitride

784


560

– interfaces

483

489

98

106

– quantum coherence – silicon

129

relaxation gettering

639

relief mechanism, interfaces

484

355

504

648 f

response functions – interfaces

465


74 f

– deep centers

192

reststrahlen bands

81

reststrahlen region

680

retarded diffusion

254

RHEED, interfaces

520

rhenium

196

rhombohedral structures

666

ribbon growth on substrate (RGS)

222

737 f

ribbon-like defects

742

rigid band model

307

rigid body translations (RBT)

384

ripening, transition metals in silicon

628

395

roughness – gallium nitride

786

– interfaces

484

– siliconoxide interfaces

525

504

457

Rutherford backscattering – extrinsic silicon

616

– interfaces

456


105

Rydberg levels – band theory

12


Index Terms

Links

Rydberg levels (Cont.): – deep centers – excitons

170

201

79

86

204 f

S s orbitals

21

σ bonding

39

σ twins

727

sapphire

775

781

satelites, interfaces

455

459

sawing

727

796

scandium impurities – silicon carbides – silicon

698 603 f

scanning capacitance microscopy (SCM)

475

scanning electron microscope (SEM)

333

scanning transmission electron microscopy (STEM)

400

scanning tunneling microscopy (STM) – gallium nitride

787


383


674

400

scattering – gallium nitride – quantum coherence

797 95 ff

Schmid law – dislocations

350

– interfaces

490

Schottky barrier – grain boundaries

422

– interfaces

532


675

Schottky barrier height (SBH) – interfaces

505

– silicides

472

522


Index Terms Schottky contacts

Links 585

Schrödinger equation – band theory

8


398


552


113

screening – deep centers

192


72

87

screw dislocations

311

347

– gallium nitride

784


668

Secco etch pits (SEP)

728

second order Raman scattering

682

secondary dislocation network

389

segregation, grain boundary-induced

431

segregation gettering

639

111

647 f

selenium – band theory

37

– silicon deep centers – zinc/cadmium compounds

204 331 f

selenium doping

265

selenium sublattice

146

selenization

762

self-assembly, quantum dot arrays

485

self-diffusion – gallium arsenide

266

– germanium

265

– point defects

242

– silicon

247

self-energy operator, band theory self-interstitial supersaturation ratio

272

52 255

self-interstitials – diffusion

233 ff


Index Terms

Links

self-interstitials (Cont.): – dislocations

366


615

– radiation-induced

246

– silicon

218

self-regulating responses, deep centers

192

self-trapped excitons

623

65

semiconductor Bloch equations (SBE)

113

sensitivity, QUANTITEM

468

seven folded types, silicide interfaces

521

shallow acceptor–donor pairs

213

shallow band-edge states

417

shallow bistability, silicon deep centers

208

shallow centers, silicon carbides

261

687 ff

shallow dislocation levels

344

shallow dislocation related states

322

shallow donors

666

shallow impurities – deep centers

170


100

shallow junction, solar cells

719

shallow levels, gallium nitride

787

shallow thermal donors (STDs)

219

shear modulus – dislocations

370

– interfaces

484

shifts, band theory

6

Shockley–Read–Hall (SRH) recombination statistics

504

– dislocations

322

– solar cells

725

Shockley partials – dislocations

298


438

– interfaces

489

313


Index Terms

Links

Shockley partials (Cont.): – quantum coherence shuffle-set dislocations

102

109

297 ff

silane

747

silane compounds

726

silane gas

546

silicide precipitates

635

silicide–silicon interfaces – epitaxial

520

– fabrication

507 f

silicides – epitaxial

472


605 ff

silicon – band theory

9f

– constrictions

301

– dangling bonds

420

– deep centers

173 ff

– diffusion

246 ff

– dislocations

309


401

– high temperature properties

597

– interfaces

495

– intrinsic point defects – partial migration – quantum coherence

190

313

362

127 ff 342 70

– solar cells

718 ff

silicon arsenides, diffusion

233 ff

silicon carbides – fundamentals

661

– gallium nitride

775


414


142


736

743


369

Index Terms

Links

silicon chloro-hydrogen compounds

726

silicon doping

265

silicon–germanium alloys

719

silicon–hydrogen bond

550

silicon oxide glasses silicon photoluminescence

274

49 323

silicon quantum dots

63

silicon self-diffusion

247

silicon self-interstitial diffusion

261

silicon–silicides growth

515

silicon–silicon bonds

549

silicon–silicon nitroxides systems

529

silicon–silicon oxides system

524

silicon tetrachloride

726

silicon wedge

469

silicon Y states

310

638

320

silver – polycrystalline compounds

756


196

– silicon interfaces

519

SIMS, interfaces

476

simulation results, grain boundaries single band effective mass approximation single interface systems

222

401 f 10 490

single particle excitations, quantum coherence

79

singularities, band theory

31

sites – band theory

9 ff

– deep centers

182


421

– interfaces

457


125


666

Slater determinant

15

149

33


Index Terms

Links

Slater transition states

194

slips, dislocations

309

solar cells

715


563

solar grade Czochralski silicon

718

solidification rates, solar cell silicon

730

solid-phase epitaxy (SPE)

507

solitons

311

354

584

357

solubility – extrinsic silicon

615 ff

– hydrogen in silicon

749

– intrinsic silicon

604

– transition metals in silicon solution growth, gallium nitride

603 ff 777

3

399

3

24

3

sp states

419

spacing, interfaces

457

spatial distributions, silicide precipitates

625

spatial frequencies, interfaces

457

spatial variations, band theory

5 ff

sp bonding sp hybrids

40

sphalerite structures – chalcopyrites

759

– dislocations

299

331

spin – dislocations

311


85

spin concentration, hydrogenated a-silicon

563

spin coupling, silicon

129

spin effects, quantum coherence

106

spin Hamiltonian

181

spin lattice relaxation, dislocations

309

spin-orbit interactions, deep centers

201


Index Terms

Links

spin-orbit splitting – gallium nitrides

789


677

spin orbitals

15

spin ordering

187

spin splitting

190

spin-valley splitting

688

687

splitting – band theory

6

64

– deep centers

180 f

190

– dislocations

340


616

– gallium nitride

789


550


148


81


677

– silicon

139

spring constants, deep centers

177

sputtering

718

square well potentials

5 ff

stability, silicon deep centers

207

687

695

723

stacking faults – dislocations

299

– gallium nitride

308 f

783 f

– germanium

440

– gettering

642

647

– point defects

239

253


729

742


632

stacking sequences, silicon carbides

328

665 ff

Staebler–Wronski effect

723

standard deformation, dislocations

309

stannium indiffusion

259

755


366

Index Terms

Links

Stark effect

114

steepest descent methods, grain boundaries

396

stick and ball structural units

392

Stillinger–Weber model – dislocations

338 f


399

Stokes shifts

130

strain, interfaces

483

strained layer superlattice (SLS)

501

413

stress – dislocations

365


390

– interfaces

490

stretched exponential decay

576

strong obstacles

360

structural dopants

303

structural properties – chalcopyrites

759


547


756


741

structural units, grain boundaries

392

401

structures – dislocations

297


383 ff

– interfaces

459

– metal-semiconductor interfaces

520

471

505

sublattice – band theory

25

– dislocations

297

– interfaces

458


146

– point defects

242

substituents, silicon carbides

666


Index Terms

Links

substitutional doping see: doping substitutional impurities

233 ff

233 f

substitutional sites – carbon precipitation

743

– deep centers

182

sulfur

204

sulfur doping

432

sulfur zinc/cadmium compounds

331 f

sun spectrum

721

supercell treatments, dislocations

334

superlattice critical thickness

493

superlattices – gallium arsenide – quantum coherence superplasticity

266

273

70 435

superposition, quantum coherence

77

supersaturation – gettering

639

– silicon precipitation

283


734

surface damage

365

surface depletion

477

surface preparation

526

surface reactions, point defects

238

surface roughening

490

surfaces, hydrogenated a-silicon

549

Sutton–Vitek SU model

394

swirl defects

253

561

280 f

swirls

728

symmetries – band theory

36 ff

– dislocations

311 f

– gallium nitride

789


553


Index Terms

Links

symmetries (Cont.): – interfaces – silicon carbides – silicon deep centers

460

483

505

674 ff

681

705

513

214

T t matrix

51

θ-function

465

tellurium – band theory

37

– silicon deep centers – zinc/cadmium compounds

204 331 f

tellurium doping

270

tellurium sublattice

148

temperature dependence, silicon carbides

706

temperature ranges – interfaces

480

– oxygen precipitation

742

template techniques

507

Tersoff model – diamond

414

– dislocations

338

tetrahedral interstitial sites

603

theoretical models, dislocations

334 ff

thermal annealing – grain boundaries

429

– interfaces

479

501

thermal donors, deep centers

204

216

thermal equilibrum – gallium arsenide

269


570

– point defects

234 ff

thermal equilibrum ionization

174

thermal expansion coefficient, gallium nitride

776

thermal fluctuations, dislocations

365


Index Terms

Links

thermally-activated diffusion

577

thermodynamics, silicon

622

thermo-ionic emission

422

thermopower, hydrogenated a-silicon

564

thickness, interfaces

468

490

thin film transistors (TFT)

546

583

585

thin films – hydrogenated a-silicon – polycrystalline compounds – solar cells

561 756 ff 718

723

750 ff

Thomas–Fermi approximation – band theory

16


87

threading dislocations

500

111

tight binding approximation (TB A) – amorphous semiconductors

46

– band theory

21 ff

– deep centers

197


554

– selenium/tellurium

204

37


670

tilt grain boundaries

393

time resolved measurements

706

time scales, quantum coherence

112

397

401

titanium – silicon deep centers

223

– solubility in silicon

607

titanium dioxide, dye-sensitized

719

titanium impurities

695

titanium interlayer mediated epitaxy (TIME)

513

titanium oxide

532

tracer self-diffusion coefficient, silicon

248

transient methods, grain boundaries

425

transistors

477

765


407

Index Terms

Links

transition elements – silicon

597


433

transition metal arsenides

520

transition metal impurities – interfaces

503

– silicon carbides – solar cell silicon

694 ff 732

transition metals – deep centers

185 f


194


746

213

222

95 ff

107

transitions – interband absorption

83 f


76

translation domain boundary (TDB)

384

translational invariance, band theory

8

translational symmetries – dislocations

313


681

transmission, dislocations

439

transmission electron microscopy (TEM) – gallium nitride

783


389

– interfaces

455

– silicon

630


735

transparent conducting oxide (TCO)

724

transport properties – amorphous semiconductors – gallium nitride

46 797


421 ff


564 ff


92


Index Terms

Links

transport properties (Cont.): – silicon carbides – solar cell silicon

704 ff 753

trapping – dislocations

303


423


566

586


133

126


701


136


747

tri-crystal growth, silicon

727

trigonal symmetry, silicon deep centers

214

TRIM programm

482

triple junction cells

586

tunneling – grain boundaries

422

– interfaces

531

twin boundaries

429

twinning

300

twins

727

twist grain boundaries

411

twisted nematic active liquid crystal displays

589

two-diode model

720

U ultralarge scale integration (ULSI)

474

ultrahigh vacuum techniques (UHV)

506

ultrashort time scales, quantum coherence

112

uncharged interstitials-substitutionals

243

514

unit cell – band theory

22

– interfaces

463


666

unterminated silicon

546

54


Index Terms UPS, band theory Urbach edges

Links 40 553

563

574

V vacancies – chalcopyrites – diffusion

759 233 ff

242

– dislocations

311

340 ff

– gallium nitride

788

– group III atoms

152

– interfaces

480


125

157

– silicon

127

132

– zinc

143

vacuum pinning

198

352

366

261

623

valence band – deep centers

170

– dislocations

304

– gallium nitride

789


417


550


320

563

72


670

677

valence force field (VFF) approach

335

339 f

valence shell electron pair repulsion (VSEPR) valence states

12

valley–orbit splitting van Hove singularities

687 31

vanadium – silicon deep centers

223

– solubility in silicon

607

vanadium impuritites

696

Varshni formula

793

velocities, dislocations

347

virtual crystal approximation (VCA)

51 f


Index Terms

Links

voids – gallium arsenide

282


125

volatile compounds, solar cells

726

volume changes, silicon precipitation

283

von Laue theorem

625 f

630 f

7

W wafers

665

670

82

86

Wannier excitons – quantum coherence – silicon carbides wave functions wave vectors

686 74 5

weak bonds – grain boundaries

417


573

weak-obstacles theory, dislocations

359

Weibull plot

527

wet cleaning

526

wetting

738

white noise interface

457

Wooten–Winer–Weaire model workstation displays

46 590

wurtzite structure – dislocations

300

– silicon carbides – gallium nitride – quantum coherence

331

665 ff 777

789

73

X X-ray diffraction

399

X-ray measurements

668

X-ray scattering (XRS)

455

X-ray spectroscopy (XPS)

34 f

40


Index Terms X-ray standing wave (XSW)

Links 521

Y Y centers

310 f

yellow luminescence

795

YLID configuration

217

Z Z-contrast imaging Zeeman spectroscopy

400

432

691 f

Zeeman splitting – deep centers

181


148

Zener formula

611

197

zero phonon line – deep centers

179


141

– silicon

130

zinc – gallium arsenide indiffusion

243


222


249

zinc compounds zinc doping

331 ff 265

276 f

zinc selenides – band theory

34

– intrinsic point defects – quantum coherence

143 73

– solar cells

756

zinc silicon gettering

653

zinc sulfides

756

zincblende pseudobinary alloys

54

zincblende structures – band theory

22 ff

– cadmium telluride

756


Index Terms

Links

zincblende structures (Cont.): – deep centers

182

– gallium nitride

789

– interfaces

460

– silicon carbides – quantum coherence

489

665 ff 73

zirconium oxide

532

zone folding

682

85


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